Do I get the miztvah of learning Torah, by listening to your answers of these peoples questions?
Actually, if you do not say a correct answer out loud and only think it, then only according to the GR"A is that considered learning Torah.
Almost all other Halachic authorities hold that learning Torah requires saying the words. Therefore you would be allowed to read the answers to these questions and even listen to a shiur (Torah class) without having said Bircas HaTorah.
So I would say that if you wish to fulfill your obligation of learning Torah every day and night, do not rely on reading or even listening to any source of Torah (mi.yodeya included). Rather, take out a sefer and review something you learned over the past year.
If one fulfills the mitzvah of Torah study with thoughts, then Torah tapes would suffice (assuming their content counts as Torah).
יחשב בלבו על שמועתו...והרי קיים והגית בו יומם ולילה; אינה אומר ודברת בו כי אם והגית בו, והגיון אינו אלא בלב
One should meditate on his studies and focus on his learning...and in so doing he has fulfilled the mitsvah of "And thou shalt meditate on them day and night". (Joshua 1:8). It does not state "And thou shalt speak it", but rather "and thou shalt meditate on it", and meditation (הגיון) is specifically mental. (Sefer al Esek HaTorah, ed. Mekhon Mar'eh p. 43).
This is also stated by Rabbenu David of Estella (13th-14th cent.):
צריך שיהיו דברי תורה על לבבו...תמיד, וזהו ההגייה יומם ולילה כי ההגיה הוא בלב
It is necessary that one have Torah matters on his mind...constantly, and this is the 'mediation day and night' (Joshua 1:8), for meditation (הגיה) is mental. (Migdal David mitsvah 4)
Ritva was also of the opinion that the mitsva of Torah study is obviously not predicated on speech. His view is particularly significant, since it is in interpretation of a passage cited by others (who fail to reference Ritva) as proof for the opposite position.
The Talmud (Berakhot 15b) is discussing whether one needs to hear the words of Shema (and whether a deaf person can therefore fulfill his obligation). R. Yosef is quoted as saying that the dispute is limited to Shema, but regarding other mitsvot there would be no question, since Deuteronomy (27:9) states: "Silence! Hear, O Israel!" The Talmud ultimately rejects the relevance of this verse, since it speaking about words of Torah; not all mitsvot:
אמר רב יוסף מחלוקת בק"ש אבל בשאר מצות דברי הכל לא יצא דכתיב הסכת ושמע ישראל...ההוא בדברי תורה כתיב
Although the relevance to the mitsvot the Talmud is discussing in negated, the verse remains potentially relevant to Torah study. What is the conclusion regrading that?
Ritva explains (there) that the point of the Talmud, is that given that the verse is speaking of Torah study, it obviously doesn't mean that literal hearing is necessary, but rather it refers to understanding:
כלו' לא בא להשמעת האוזן אלא לכונת הלב לשינון והיינו בהרהור
That is, [the verse] didn't mean to connote hearing of the ear, but rather the heart understanding the study, which is through thought.
This is in turn cited by the Shitta Mekubetset there.
This also appears to be the view of the Meiri (as noted here). The Mishna in Berakhot (3:4) states that a baal keri should think the Shema, and not recite the blessings for it. The Meiri (Berakhot 21a) asks that the Shema is Torah, and its recitation ought to require a blessing before, so how can the Mishna exempt the baal keri from reciting the blessings before the Shema, which ought to be necessary for the mitsvah:
יש שואלים ק"ש תורה היא והיאך אמרו במשנתנו שאינו מברך לפניה אף בהרהור והרי ברכה לפניה מן התורה
Clearly, he assumes that thinking words of Torah should require the blessing just as speaking it does. Presumably, this is because he assumed it fulfills the mitsvah (like the Aharonim we will quote below, who use this argument).
This is also the view of R. Yaakov Emden who writes in a responsum (Sh'elat Yaavets (2:140) who writes that one must make a blessing before thinking Torah, since that too is a mitsvah, as the verse states "And thou shalt mediate on it". (It should be noted that there are reasons to not recite a blessing, even if it is a mitsvah, but that discussion belongs elsewhere).
This is also the view of the Vilna Gaon in his glosses to the Shulhan Arukh (OH 47) where he questions the ruling that one does not recite the blessings upon merely thinking Torah, on the grounds that thoughts too fulfill the mitsva of Torah study, as indicated by the verse:
צ"ע דכאן מברך על המצוה וכי ליכא מצוה בהרהור והלא נאמר והגית בו וכו' ר"ל בלב כמ"ש והגיון לבי
See also Yeshuot Yaakov (47), and Shut Orah Mishpat (OH 11) in addition to Yabia Omer (Vol. 4 OH 8:15) for a listing of many more who agree. This is also the assumption of Rav Soloveitchik (Reshimot Shiurim to Berakhot: Ha'arot to 11b).
Significantly, some understand this to be Rambam's view as well.
Rambam writes in Hil. Talmud Torah 3:14 that learning out loud facilitates memorisation, whereas silent learning breeds forgetfulness.
כָל הַמַּשְׁמִיעַ קוֹלוֹ בְּשָׁעַת תַּלְמוּדוֹ, תַּלְמוּדוֹ מִתְקַיֵּם בְּיָדוֹ; אֲבָל הַקּוֹרֶא בְּלַחַשׁ, בִּמְהֵרָה הוּא שׁוֹכֵחַ
This strongly implies that one does fulfill ones obligation to study Torah, for if not he should have said the bigger issue; that one doesn't fulfill the mitzva, rather than just saying that one is likely to forget.
R. Nachum Rabinovitch author of the monumental Yad Peshutta commentary to the Rambam, concurred with this inference.
I have found practically no Rishonim who dispute this explicitly. For a couple of Rishonim who might disagree, see here. This is similarly the perspective of R. Gedalya Drexler who writes:
וברור לפי"ז שהבנת הראשונים ז"ל היתה שאין ת"ת מן המצות התלויות בדיבור
It is clear according to this that according to the Rishonim, talmud Torah is not a mitsva dependent on speech. (Kovets Beit Aharon V'Yisrael (83) p. 45.
However, some Aharonim such as the Pnei Yehoshua (Berakhot 15b who disagrees with Ritva above) do dispute this ruling.
According to the Shulchan Aruch Harav (Hilchos Talmud Torah ch. 2:12).
וכל אדם צריך ליזהר להוציא בשפתיו ולהשמיע לאזניו כל מה שלומד בין במקרא משנה ותלמוד אא"כ בשעת עיון להבין דבר מתוך דבר. וכל מה שלומד בהרהור לבד ואפשר לו להוציא בשפתיו ואינו מוציא אינו יוצא (ברכות *ב׳ ע״ש בפ״י ד׳ ט״ו) בלימוד זה י״ח מצות *ולמדתם אותם וכמ״ש לא ימוש ספר התורה הזה מפיך והגית בו וגו׳. וכמו בכל המצות התלויות בדבור *שאינו יוצא בהן י״ח בהרהור אא״כ שומע מפי המדבר *שהשומע כעונה בפיו. אך אם מוציא בשפתיו אף על פי שאינו מבין אפי׳ פירוש המלות מפני שהוא ע״ה ה״ז מקיים מצות ולמדתם. ולפיכך כל ע״ה מברך ברכת התורה בשחר *לפני הפסוקים. וכן *כשעולה לס״ת.
Everyone must take care to say with their lips and hear with his ears whatever he learns, whether it be Psukim, Mishna or Gemara, unless he's learning in depth. And whenever one learns quietly and doesn't say his learning out loud doesn't fulfill his obligation to "learn it" ... Like any other commandments which are dependent with speech, one doesn't fulfill his obligation through thought, but has to hear it from someone else.
But if one says the words and doesn't understand them, he fulfilled his obligations 
His source is the Gemara in Berachos Berachos 15b which says
מיתיבי לא יברך אדם ברכת המזון בלבו ואם בירך יצא אלא אי אתמר הכי אתמר אמר רב יוסף מחלוקת בק"ש דכתיב שמע ישראל אבל בשאר מצות דברי הכל יצא והכתיב הסכת ושמע ישראל ההוא בדברי תורה כתיב: The Gemara objects based on what was taught in a baraita: One may not recite the Grace after Meals in his heart, inaudibly, and if he recited the blessing in that manner, he fulfilled his obligation. In this example of the rest of the mitzvot, the obligation to hear the recitation of the blessing is only ab initio. Rather, Rav Yosef’s statement must be emended. If this was said, it was said as follows; Rav Yosef said: The dispute as to whether or not a deaf person fulfills his obligation is only in the case of the recitation of Shema, as it is written: “Hear, Israel.” But regarding the rest of the mitzvot, all agree that a deaf-mute fulfills his obligation. The Gemara asks: Isn’t it written: “Pay attention, and hear, Israel”? The Gemara responds: That verse is written with regard to matters of Torah; one must pay close attention to what is written in the Torah.
Which implies that Shema Yisrael (implying one has to read it out loud) apply more to Torah study than to Blessings (about which the Gemara debates earlier).
So too, the Shev Yaakov says that since the commandment of Learning Torah comes from "Veshinantam Levanecha" - teaching it to one's son, the only way to fulfill the Mitzvah is in a way that a hypothetical son would be able to understand, and must be done audibly.
Also, the Shulchan Aruch (47:4) and Chayey Adam say that one doesn't say blessings on the Torah if one just thinks Torah thoughts (though the Gra does ask there that Higisa implies thought, not speech, and the Chayey Adam says that the Gra leaned toward saying Birkas HaTorah on thought), implying that one doesn't perform the commandment of Torah study through thought .
So too, Rabbi Braun explicitly says that one does not fulfill his obligation of learning Torah through hearing tapes. Though he writes that there are some opinions who say that silent reading from books isn't considered "thinking", the Shulchan Aruch HaRav and the Rama Orach Chaim 67 (Which says that one is allowed to silently read from Seforim when he is holding in a place where one is not allowed to interrupt. If silently reading from a text is considered speech, then reading would be the same hefsek as speaking).
 The footnote there says that this is a typo.
 The next Syif says that this is only true about the written Torah. When one learns from the Oral Torah one doesn't fulfill his obligation unless he understands what he says.
 Though the Shaagas Aryeh Siman 24 implies that while one fulfills the Mitzvah of Torah study through thought, one can't say the blessing over it since the source-verse for the blessing is "Ki Shem Hashem Ekrah", implying that there must be words spoken.
 In the Maarei Mekomos tab.
 Though not recommended, as he may absentmindedly start reading out loud.