How do we explain our belief in the story of Yonah living in a Fish, the Splitting of a sea, and all the other fantastic Miracles described in the Torah to a person who was not brought up religious; and maybe in fact does not believe in religion because of these parts of the Torah? also is saying the story of Jonah Allegorical heresy?
I think it is fruitless to try to prove torah to non-believers, though people try (e.g. the "torah codes" folks looking for hidden messages in the text). Faith is not science.
As for explaining our belief in the truth of torah, one can ask: Is there anything that Ha-Kadosh Baruch Hu cannot do? The torah describes miracles, which were either performed on the spot by God (the pshat) or programmed into the creation of the world (Rambam, I believe, but no cite). God can act in the world; this doesn't obligate Him to do so in the future ("why don't we see giant fish swallowing people nowadays?"), but it also does not preclude it (e.g. resurrection).
As I understand it, Maimonides in his Guide to the Perplexed mentioned that some Jewish philosophers had gone so far as to say the very existence of Biblical characters such as Abraham was allegorical -- that's going too far. Maimonides himself interprets certain episodes of angelic interaction in the Torah as dreams, allegories, or the like, which (unsurprisingly) ruffles some traditionalist feathers.
My understanding is that the standard, normative understanding of Jewish faith would be that the Jonah story actually occurred. I think, and I could be wrong here, that if someone chose to understand it as allegory or a dream or the like, however, Jewish Law would not treat him/her as a heretic (koifer in your question), as their belief would not be in violation of the Thirteen Principles of Faith (while "I believe that all the words of the prophet are true"; a dream or allegory could be "truth", I presume); just as the rabbis of the Talmud themselves debated whether Ezekiel's vision of the Dry Bones was a physical reality, or just something seen in a prophetic trance. (Though Ezekiel opens the story with a prophetic-vision experience...)
To agree with Monica, I don't think there is going to be the a-ha, gotcha! that's going to do it for an atheist, certainly with regards to a story like this. We believe that G-d can do anything, including bending the laws of nature as needed.
A more interesting question is to transcend the how and ask why (as Lord Sacks likes to say, science is the how, religion is the why): why the whale (or "great fish", whatever it was)? As I read it, Jonah's job was a prophet, and he had to go warn Nineveh. Now he had all sorts of reasons why he didn't want to do his job -- Nineveh was the capital of Assyria, which would go on to attack Israel, not to mention make us look awfully bad that we ignored all our prophets' calls to repent). But that's not our place. G-d is G-d, and we are humans. Once Jonah started by breaking that order, he starts to see the laws of physics go topsy-turvy around him. This explains the epilogue to the book that the Midrash fills in, where Jonah bows down and says, "Okay G-d now I get it; the world is Yours and You do with it as You wish."
How to explain to an atheist? Don't look to Johnny:
We don't change the Torah or our religion in order to make it more palatable to atheists. If they don't believe it because of miracle X, and they do not believe in miracles, then that is fine. If you deliberately strip out all the miracles, such that Moses builds a pontoon bridge to cross the Red Sea, then you are left without religion or belief in God's intervention in the world anyway.
That is not to say that other concerns should not lead to a naturalistic explanation of Biblical miracles. Various Rishonim were led by their philosophic beliefs to believe that miracles which violated the natural order would reflect a flaw in the original plan of creation, and thus tended to explain miracles in natural fashion. (See e.g. Ibn Ezra.)
And there is a rather strong basis in the Biblical verses to say that God manipulated nature. For instance, see how the wind dried up the waters in the Reed Sea. Or how the wind carried the locust plague in Egypt, where we know from observation nowadays that locusts regularly migrate along that path.
And I would be extremely willing to say that Yonah is metaphor. Not because of the big fish part. But because as a student of literature, Yonah reads to me like a morality play and a lesson about the importance and nature of repentance, and theologically why God allows it. The prophet delivering the message, the sailors, the city of Ninveh, and even the animals all repent. I think Chazal possibly saw it as metaphor as well.
But don't water it down just to make it palatable for an atheist.
As to your question of explaining miracles: You don't. It is quite logical that if an omnipotent G-d wanted our people to know of His interaction with humanity, He would have to display this clearly at least once, subsequently being passed down to future generations. (Yonah was a private miracle, only publicized later. I don't know the purpose of the miracle, buy perhaps it is for the remez that can be learned.)
This person has a fundamental issue with an Omnipotent G-d.
A kofer? It would seem not. The Seforno Breishis 3:1 learns that there was no Nachash in Gan Eden, but that "nachash" is the Yetzer Harah and the verbal conversation never took place.
Although belief in miracles is an important belief in Judaism, I do not think it is kefirah to believe that the laws of nature are never broken, even if it is not the traditional Jewish view. If a person believes God created absolute rules that He never breaks, its probably OK, as long as he accepts that there is some form of Divine Guidance in this world.
There were some scholars who tried to give rational explanations for many miracles. For example, the Ralbag often tries to explain how certain miracles didn't violate the laws of nature (Although he's not exactly a fully traditional source).
There were no physical laws necessary for the sea to split, just very large unlikelihoods. Similarly, many miracles can be just explained as statistically unlikely quantum events. This is acceptable, for clearly God is behind the events. Some may wish to go further and say the splitting of the sea was just a very low tide or something. That is probably not heretical either. Those who do not accept Judaism may find such that explanation more reasonable.
As for the story of Yonah, there were those (possibly the Vilna Gaon) who explained it allegorically.
I think you need to first reconcile science with judaism first to the person, before you explain the question of miracles, as this is the underlying question.
In Michtav M'Eliyahu (Vol.2 pg 264), Rabbi Dessler discusses the miraculous account in the book of Yehoshua, whereby, the Jews were instructed to blow shofars and the huge walls of Jericho came crumbling down. He says the following: "see the Ralbag commentary who pointed out that there is grounds for mistakenly thinking that the blowing of the shofar is what brought down the walls, since the loud noise could produce air movements which in turn cause vibrations in the wall. In truth, whoever wants to deny can explain away every miracle with remote explanations, and crookedly and stubbornly claim that it was an accident. And we have already explained that when G-d performs a miracle, He leaves open an escape to allow for erroneous explanations. This he does for the stubborn person in order to not diminish the possibility of free will." (another example of this: After Samson lost his miraculous strength, the verse says: "the hair of his head began to grow again after he was shaven" (Shoftim 16:22) which the stubborn person can use to mistakenly attribute his strength not to a miracle of G-d but rather to his hair, despite that a few verses later he prayed to G-d saying (ibid verse 28): "strengthen me, I pray You, only this once more, O G-d, that I may be this once avenged of the Philistines for my two eyes".
The Ramban commentary on the torah (Exodus 12:16) says: "...for man has no portion in the Torah of Moshe Rabbeinu unless he believes that all our matters and happenings are completely miraculous, there is no nature or ways of the world.."
The universe around us, and especially life is miraculous. Only that G-d instituted a natural order so that there would be room for free will, namely that the stubborn person who wants to shirk his religious duties can find an escape by attributing everything to "mother nature". If we could find even one phenomena that could not be explained scientifically but could only be attributed to an open miracle of G-d, and this phenomena could be reproduced and demonstrated to everyone on demand, then free will would be null and void, since nobody could deny the existence of G-d and everything this entails (morality, religious service, etc.). This is why G-d created the world in such a way that everything in it can be explained through science. Life has the capability of being attributed to evolution. The brain can be experimented with and studied with the same physical laws which explain the motion of electrons in a copper wire. Astronomers who study the universe, will find they can attribute its existence to a Big Bang billions of years ago and study that event with natural laws. Physicists can probe the atom and find ever more exotic particles and explain their behavior through scientific theories. All phenomena has the ability to be attributed to the "non-miraculous". This is a necessary side to creation in order to preserve man's free will.
However, the thinking person will see that there are big holes in the scientific perspective. The Big Bang has the big question of "what caused it?". Life has the great question of "how can something so complex occur by mere trial and error"? The true truth-seeker will be bothered by such things. And even the inanimate world reflects an infinite wisdom as the nobel prize winning physicist Richard Feynman, considered one of the most profound physicists of all time, wondered:
"It always bothers me that according to the laws of physics as we understand them today, it takes a computing machine an infinite number of logical operations to figure out what goes on in no matter how tiny a region of space, and no matter how tiny a region of time. How can all that be going on in that tiny space? Why should it take an infinite amount of logic to figure out what one tiny piece of space/time is going to do? So I have often made the hypothesis ultimately physics will not require a mathematical statement, that in the end the machinery will be revealed and the laws will turn out to be simple, like the chequer board with all its apparent complexities. But this is just speculation." (from his book: The Character of Physical Law - Chapter 2 - the relation of mathematics to physics)"
So really there is no contradiction between the torah and science. There must be free will, and in order to preserve free will, G-d must have a way of allowing the stubborn person to attribute everything in the universe to the non-miraculous, i.e. science. The more we discover, the more we will find scientific explanations for everything. In earlier days, when people thought the world was flat, the ambiguity was "what's beyond the earth?" and that was enough ambiguity to balance free will. Now as technology discovers more and more, so too G-d must reveal deeper reasons for phenomena such as the ever more exotic laws of quantum mechanics or the mind boggling complexity of biological phenomena, so that no "open miracles" can be found which would nullify free will. )
(now to answer the question:) This is not to say that open miracles never happen. The Tov Halevanon commentary in Chovos Halevavos gate 4 ch.4 says: "G-d does not do public miracles without great necessity". Sometimes, G-d will come out from behind the mask of nature and perform miracles, but only if there is a great necessity since this interferes with His plan of giving man free will. For example, the ten plagues in Egypt were necessary because they were the prelude to the giving of the torah, and there had to be no doubt that the torah was of Divine origin (Drashas HaRan). Or, Avraham's being thrown into a blazing furnace and surviving was necessary since he would be the father of the Jewish people.
I agree very much with Monica; in my experience it is generally not a great idea to discuss beliefs with people in general and particularly not in this situation.
I was raised in a Christian home to a very religious family (missionary relatives, church every Sunday, etc.) and am now an atheist. I left my parent's religion because I have spent most of my adult life studying science and the logic of science is more appealing to me than the illogic of religion. Nonetheless, and I expect I'll get a lot of flack for saying so, I am converting to Judaism.
It is true that I don't "believe" literally the stories told in the Torah, but that actually does very little to my impression of Judaism as a whole. It seems to me that people who are raised and stay in a religion their whole life see it as a set of beliefs where people outside of the group see it as a set of practices. This would explain why, when violence is committed in the name of (any) religion, people of that religion say "They aren't really (insert religion of choice here), they just say they are." whereas people outside of that religion say "Wow, look how violent (insert same religion here) is!"
In a situation in which an atheist raises your beliefs, they are likely trying to get you to evaluate them from an outside perspective. If you are comfortable with doing this, the best policy is to be honest. On the other hand, if you happen to be talking to an atheist and no one mentions beliefs, don't bring it up; instead, try to focus on what the practices of the religion mean to you.
I will try to answer the beginning of your question: "Answering Atheist". For the second part "about Torah miracles" there are already dozens of good answers. I will only try my 2 cents.
In this answer I will assume you care very much about the person you are talking to and you both care genuinely about finding the truth. If this is so you two can try to find the truth together, search books, argue, and so on... From this type of discussion you can gain very much. If this is not so, maybe it is pointless to argue, as Monica said.
I think, first of all, you should seek your answer to those questions. What do you think constitutes a miracle? Do you believe those miracles are true? And why? If you feel you are unsure about your own positions think deeply about them. You should not think in defensive ways, you should strive for the truth. If he is right, he is right, but before you act consequently (maybe the Torah is not right, after all), you should be sure that he is right.
A good idea is to go to a Rav that can handle this type of questions (Torah vs Science) and to bring your friend with you. Then ask the Rav what is the answer to the atheist's claims. If he cares about the truth, he can gain from speaking with the Rav, and you, too, can learn very much.
A Rav is only one of the resources you can find and it has not to be a Rav, it can be someone with enough familiarity with the topic. You can search proofs on your own (many are provided as answers to this question) that help you to defend and strengthen your position.
I feel somewhat obliged to share my position about the topic. I believe in spirituality and I feel a connection with Hashem. I know that he answers my prayers, for example. So spirituality can influence reality. I cannot explain how, but I know for sure that if Hashem wants something, he obtains the result. From a religious perspective it doesn't matter very much how. I happen to have a degree in Physics, and from what I learned, I know that there are "principles" or "fundamental theories" that are universally believed to be true because no experiment ever contradicted them in a fundamental way (e.g. conservation of mass/energy, uncertainty principle, the fact that a "theory of everything" should have a flavor of QFT and involve many fundamental symmetries thus implicating conservation of many quantities by Noether's theorem but I am really digressing now...). My position is that: (1) it can well be that a miracle produces something that contradicts these natural laws (2) there are many ways that almost all of these natural laws can remain true. I can myself come with some (what if we take quantum perturbations of slim probability into account? or, more simply, some geological or biological curiosity? there are so many thing that are possible but don't usually happen!). You can also find that the Torah itself, when interpreted through a Midrash or just a small insightful consideration, can change its face (maybe the "taninim" in the Genesis are dinosaurs, or Adam had relationships with hominids...). It doesn't make much sense to stick with one "natural theory" and one interpretation of the Torah and make them collide!