Can someone be chasidish (or a chasid) if they don't dress in a distinctively "Jewish" way or don't keep all the minhagim such as "cholov yisrael", saying tehillim on rosh chodesh, etc? Does it depend on which chasidus? In other words, is chasidus mainly in the "philosophy" and internal thoughts and beliefs or is it mainly external things like dress and minhagim, or are both required?
I believe in the past there have been Hassidic thinkers who did not adopt the garb, but today we usually associate the two (but not necessarily). (Was it the Kotzker who was a Hassidic thinker but not dresser?)
Rabbi Dr. Moshe Dovid Tendler recalls an interesting conversation he had with Rabbi Menachem Mendel Schneurson, the late Lubavitcher Rebbe zt'l:
I asked him, “Why is it that your sheluchim (emissaries) take a fellow and make a Chasid out of him by putting a kapote (long black coat) and hat on him – even though he knows nothing at all about Judaism?” The Rebbe answered me, “But it works this way. If we tried it any other way, it would not work.” A person has to know that he belongs to the rest of his community before he can actually become part of it, and these outward signs allow him to do so.
You could read into that what you will; I'd read into it that for the masses, adopting a complete brand is most effective. But if some hybrid works for you, why not? (The Rebbe didn't answer "because keeping kosher is useless unless you wear a black hat!")
As for what to keep, I can refer you to Rabbi Moshe Feinstein's responsum whether someone who has worn traditional Polish Jewish garb who moves to America must maintain it; he writes that you should keep the minhag (customs) of your ancestors, but not their chumras (choices to be extra-strict); for instance (I would gather), if you accept R' Moshe's logic that chalav yisrael in America today is a chumra, then you wouldn't be bound by your family's adoption of that chumra. Minhag would be more like whether to stand or sit for kiddush. He also writes that in his (pointedly non-Hassidic opinion), the evolution of Hassidic garb was that a great rabbi chose to wear a long garment (considered more modest, and the sign of a scholar) and make it out of silk (to avoid prohibited mixtures of wool and linen); it wasn't that the rabbi made an enactment that his followers should follow his example, just that they chose to emulate him on their own. (Rabbi Feinstein further argues that American clothing today is "American Jewish garb" and no worse than "Polish Jewish garb.")
I'm sure there are still Hassidic groups that operate with very strict definitions of "in" and "out", but in today's world, more and more people find themselves in different ways; if the way that works for you to believe in G-d and Torah and keep 613 is by using Belz tunes, Lubavitch philosophy, and Lithuanian customs, at the end of the day I'd stay you're still a good Jew. (But that's my opinion.)
Now if you choose to hybridize legitimate streams of Orthodoxy, be aware that some people will be more okay with this than others, that it might be worth thinking about what practices to do publicly vs. privately, given your environment; and that some bridges are best not burned. That's a path you have to find for yourself.
Much hatzlacha (success) in your journey!
Chasidus strongly believes in keeping the Minhagim and dress of the previous generations. Thus someone that would not keep the Minhagim is not really being Chasidish. However there are different Chasidishe groups that dress in different manners (Lubavitch wears bend down hats, Stolin wears regular brim up hats, Bobov wears high beaver hats, Satmar wears low beaver hats). Thus you can be Chasidish in many different modes of dress if you follow the dictates of that particular Chasidus.
My father always tells me that when the Torah talks about a murderer it is talking about a Frum Jew that has murdered someone. He does everything a Frum Jew does yet he went ahead and murdered someone. Thus there are people that may consider themselves Chasidish even if they are not doing everything 100% right.
A person can be a Hasid without necessarily belonging to a specific group. In my experience and from what I heard in the name of many Rebbe's, some of the greatest Hasidim they knew did not dress the part and did not blindly follow whatever the other Hasidim were doing. It does not mean that one can just pick and choose, rather one should be a thinker and adopt those practices that will help him in his service of Hashem and not adopt those practices which will hinder his service of Hashem.
I have seen many Chabad wearing only the "normal"clothing. The ones at Ben Gurion asking if you wish to lay Tefillin are a good example. It is also not uncommon in The Netherlands.
I think that Breslov is "mainly in the philosophy" as you say. Though Breslovers often have distinctive dress, on the whole they're much more varied than other branches of chassidim. As far as minhagim, the central points that one is supposed to adopt are very private (spending Rosh HaShanah in Uman is not so private, but the other two central pillears, shemirat haberit and hitbodedut, are very private).
On the Lubavitcher Rebbe's 70th birthday, he was asked by a college student if it was possible to be his Chassid without donning the Chassidic garb, without growing a beard. The Rebbe replied, "Every day, even now, I wake up each morning seeking to make it better than the day before. If you only make the commitment to do this, to consistently add in goodness, I will be proud to call you my Chassid." (source)
This is the position of the holiest, and it is surprisingly complex. (Note that the Rebbe does not say one need not don the beard or lavush.) In the longest view, the Rebbe is exactly correct, and there could, in my judgment, be no error in living by these words.
You may encounter some earthly resistance if you try to become a chosid "only" in this way. The fact is that, although you may indeed be a chosid, many people will not consider you a chosid if you don't do some (many) basic and not-basic things that have come to be associated with chassidishkeit. Just some of these are davening the nusach; learning the chassidus daily; keeping the (extensive and detailed) minhagim; being machmir on halacha; following the directives of your Rebbe; cultivating hiskashrus to your Rebbe; wearing the lavush 100% of the time, including variations for sleeping and in the home and for sports/recreation; living in a Chasidic community (without which, you'll find, the rest is almost impossible); following a rov from your community; taking part in farbrengens and tishn; keeping the kashrus of your chasidus (it can be hard); speaking Yiddish (not hard); marrying a chasidish wife and raising your kids in the chassidus; sending your kids to chasidish school (not a lot of secular education there); growing a beard and peyos; going to the mikvah, sometimes every day; not associating socially with the opposite sex; with limited exceptions, not listening to secular music nor going to secular school nor reading secular books nor watching movies nor reading secular news; rejecting a certain amount of technology; and this is only the tip of the iceberg.
"Dancing at Two Chasunas"
If you don't do all or most of these things, you will likely find that others in the chasidus do not really consider you to be one of their own (although they may anyway be kind and supportive). This may start to be a problem when you are looking for a shidduch for yourself or for your children, or looking to send your children to school. For these things, it is usually not enough to be simply "very frum" or to have great respect and love for the chassidus and a wish to connect. These communities survive to a significant extent by rejecting outside influence, and, though they may have some interface with the outside world for purposes of kiruv, politics or business, the purely Jewish, chasidic qualities of their homes and institutions are precious to them; if you wish to join these homes or institutions, you need to be committed to the same. (Or have--sometimes--a great deal of money which you intend to spend your life giving to the community.)
Permissible Variations and Alternatives
Some variations are permissible within chassidic communities. Many communities have both a more modern or idealistic element and a more traditional, conservative, "chassidish" element. Sometimes these elements live in different towns or on different streets within a town. Usually, the first group will permit more flexibility in the dress and lifestyle, as well as in politics and in interaction with the outside world. The second group is internally considered more prestigious, may look down on the other group and will not marry them, and may follow a stricter minhag considered closer to the Rebbe's own; this group may consist of a bifurcated mix of families with yichus in the chassidus, and newer families/individuals who are trying to accrue some chassidic capital in compensation. (This strategy works and is necessary in some cases and not others.) The first group may include eyniklakh of the second group who seek more freedom within the chassidus, and rely on their family status for continued inclusion, as well as a certain proportion of baalei teshuva. Moving between the groups is possible, but easier in one direction than the other.
Outside of the traditional chassiduses are some non-mainstream options such as neo-chasidism, heimishkeit, chasidei klali, Carlebach, yishuvim, and probably many others I've never heard of. In some Chasidic communities--well, in Crown Heights--there are groups of young Modern Orthodox who live on the periphery of the community and daven at the community's most liberal shuls and at their own shuls. In Israel, there may be a great deal more flexibility in defining chasidishkeit; I have heard of a number of organizations and communities where a more fluid, liberal notion of chasidishkeit is embraced, possibly in consort with a hippie-style and/or "return to the land" lifestyle. Breslov seems to have a particularly sizable contingent of such types. Seekers in general are welcomed in Israel and may have luck finding their place.
Why be a Chosid?
As for why all this is worth it -- a Chasidic community has many benefits. First of all, you will, in fact, live for G-d. It is very difficult to achieve this thing in other environments. It may be that doing so is infinitely important. If that is worth something to you, then it may be correct for you to become a chasid.
Second, you will live an old, dignified, and beautiful Jewish life every day. This is a life charged with immediacy and poignancy. It is charged with the past -- the entire weight of the past -- and of the miracle of its survival. The culture itself is a remnant of a precious otherworld that was nearly obliterated from existence, and that survives only by virtue of the efforts of the few and proud to resuscitate it with all their might again and again at every moment. If you become a chosid, you will become part of this resuscitation and part of this survival. This itself is an acutely meaningful thing.
Third, the lack of freedom in Chasidic life is repaid with a deep sense of security. There are no questions about what to do: the generations before you have already determined what is worth doing. You do that. You trust that it is enough for a good life. If you have a question, you ask your Rebbe, and your Rebbe will answer you with complete wisdom. The important things in life--family, community, physical integrity and dignity, peoplehood, and devotion to G-d--become your rights; if you struggle with any of them, others will bend over backwards to help you, because they know that these things are essential things for the soul.
How to Join a Chasidus
If you decide that the hardships are worth it to you, and you do want to join an "official" chasidus, here are my recommendations. First of all, find a Rebbe and a community. These may require research and much trial and error. Then move there, find a job there, and enroll in a yeshiva there. Once you are a bit established, you can look for a shidduch there, if need be, and start a family. No matter how closely you conform to the ways of the chassidus, you will spend your life as a baal teshuvah, and will always be seen as "second-class" in many ways. Still, if you can manage, you will be rewarded with all the gifts I have described. And your children will be luckier; they and especially their children will have chasidus as their birthright.
Being a chasid is not an easy road, but it may be the right one (for some) for reasons of this world and the next. Think about it carefully and do it slowly and with self-compassion. Much hatzlacha and much bracha, seriously.
As noted in another answer the Lubavitcher Rebbe told a young man who once came to him and said I would like to be your chossid but don't see myself donning a sertuk as you wake up every day with the commitment to be better than the day before I will be proud to call you my chossid.
"Chabad, however, the emphasis was not so much on how one looked on the outside, nor on creating an insular community by dressing differently. Rather, the emphasis was on creating the internal strength and warmth to withstand assimilation, through studying and internalizing the teachings of Chassidism.
It is this internal warmth which gives Chabad emissaries throughout the world the strength and warmth to go out to remote communities where there is no Judaism. Not only do they have no fear of being assimilated, but on the contrary, they serve as a beacon of light, bringing the warmth of Judaism to the remotest of regions. In the end, while garments have importance, it is our actions and convictions that truly define us." http://www.chabad.org/library/article_cdo/aid/3755339/jewish/Why-Do-Many-Chassidim-Wear-Shtreimels-Fur-Hats.htm#utm_source=facebook&utm_medium=social&utm_campaign=fb_en
To be a chossid is much more than clothing. The clothing is only useful as a reflection of the inner. Simply donning a set of clothing will not transform you. Someone once asked the Rebbe Rashab during a farbrengen what it meant to be a chossid... http://www.chabad.org/library/article_cdo/aid/2183/jewish/The-Lamplighter.htm
Here is a link that is an audio explaining a maamar by the Lubavitcher Rebbe about what is the purpose of chassidus... http://www.chabad.org/multimedia/media_cdo/aid/1036483/jewish/On-the-Essence-of-Chassidus.htm