Q: It is correct to say today that we and Yeshivat Ha-Gush are two separate worlds, just as one would say about us and the Charedim?
A: I don't know who "we" is. All the Zionist Yeshivot are the same, with minimal differences. And the same with the Charedi world: that which we share is far beyond that which separates us. It is forbidden to create sub-groups within the Nation "And who is like Your Nation Yisrael, one Nation in the Land" (see Maran Ha-Rav Kook's article "Masa Ha-Machanot" in Maamrei Ha-Re'eiyah).
Q: When people are arguing and someone calls the other a Nazi, how should he respond?
A: He should not respond. By stooping to this level, the name-caller disqualified himself and lost the argument. This is called "Reductio ad Hitlerum" or "argumentum ad Hitlerum", i.e. where someone compares his opponent's views with those of Adolf Hitler or the Nazi Party (see Wikipedia for this expression).
Q: Is it worthwhile to have a "Ta'anit Dibur" (a fast from speaking)?
A: It is preferable to have an internet fast.
Q: What is the blessing on papaya?
A: Adamah. Shut Yechaveh Daat (4:52).
Woman Covering Hair during Davening
Q: If a married woman Davens at home on her own, is she obligated to cover her hair?
A: The basic Halachah is that she is not obligated, but in practice, the Minhag is for a married woman to cover her hair when Davening, even if she is alone. Shut Yabia Omer (Vol. 6 Orach Chaim #15). Sefer Tefillah Ke-Hilchata (Chap. 7 note #24 in the name of Ha-Rav Shlomo Zalman Auerbach).
Revelation of Eliyahu
Q: Why did Eliyahu Ha-Navi appear to the Arizal and the Vilna Gaon but not the Rambam?
A: The Rambam had a revelation of Eliyahu through his intellect.
Kashering and Toveling
Q: If one purchases a non-Kosher utensil from a non-Jew, which should be done first – Kashering it or immersing it in a Mikvah?
A: Kashering it. Shulchan Aruch, Yoreh Deah 120:2 (And see a discussion of this issue in Ha-Rav's commentary on the Kitzur Shulchan Aruch 37:4).
Woman who is in Doubt about Birkat Ha-Mazon
Q: I am a woman and I am almost certain that I did not recite Birkat Ha-Mazon but I am not 100% sure. What should I do?
A: There is a huge debate regarding whether a woman who is in doubt about whether she recited Birkat Ha-Mazon should recite it or not. She therefore should not recite it (Piskei Teshuvot 186:2). But if you are almost certain that you did not recite it, then you should recite it.
Writing on Fog
Q: It permissible on Shabbat to write on a window which is fogged up?
A: No (And so too Shemirat Shabbat Ke-Hilchata 16:28).