Q: Is it possible for a fast day to fall on a Friday?
A: Asara Be-Tevet is the only one. Shulchan Aruch, Orach Chaim 428, 550.
Q: Is it permissible to eat before Alot Ha-Shachar?
A: Yes, if one made such a condition before going to sleep, and it is sufficient to think it (Shulchan Aruch, Orach Chaim 564:1. Mishnah Berurah #3. Sha'ar Ha-Tziyun #3).
Q: It is permissible to brush one's teeth?
A: Yes (Shut Minchat Yitzchak 4:109. Yalkut Yosef – Moadim, p. 534).
Q: To bathe?
A: Even one who is strict not to bathe on Asara Be-Tevet may do so when it falls on Friday in honor of Shabbat (Mishnah Berurah 550:6. Shut Yechaveh Daat 5:41).
Q: Can someone who is not fasting serve as Shaliach Tzibur?
A: No. Shulchan Aruch 566:2.
Q: It there a Minhag to give Tzedakah?
A: Yes. The reward of a fast is the Tzedakah given. Berachot 6b. And there are those who give the amount of food they would have eaten. Mishnah Berurah 566:12.
Q: Is it permissible to taste the food in honor of Shabbat if there is a concern that the taste will be spoiled?
A: Taste a tiny amount and spit it out. Rama, Orach Chaim 557:1. Mishnah Berurah #6.
Q: Should one also put on Talit and Tefillin at Minchah?
A: The majority do not put on Tefillin at Minchah. Sha'arei Teshuvah 34:2, 37:1. Biur Halachah 37 d.h. Be-Sha'at. Regarding Talit, there is no problem to do so.
Q: If the community Davens after sundown, do they still read the Torah reading of Vayechal and say Anenu in the Shemoneh Esrei?
A: Yes. But they should take care to Daven before sundown since there is a belittling of the honor of Shabbat in doing so (Shut Dvar Yehoshua 3:63. Piskei Teshuvot 550:10).
Q: Do we recite Birchat Cohanim at Minchah (in Eretz Yisrael)?
A: Some say in the name of the Chazon Ish that a Cohain who is fasting recites Birchat Cohanim even at Minchah Gedolah, and this is how they act in his Beit Midrash in Bnei Brak (Kehilat Yaakov Bereachot #29. Maaseh Ish Vol. 2, p. 98), and some also follow his ruling in other places (Moadim Uzmanim 7:248). But this is an innovation. In general, we only recite Birchat Cohanim at Minchah Ketana (Shut Shevet Ha-Levi 8:23. Hilchot Shlomo, Chapter 10 note 21). And some say only after Pelag Ha-Minchah (Luach Eretz Yisrael). See Piskei Teshuvot (128:1) and Ishei Yisrael (Chapter 27 note #116).
Q: Do we recite Vidui and Tachanun at Minchah?
A: No (Mishnah Berurah 550:11). And the same is true of Avinu Malchenu (Mishnah Berurah 602:3), and some Sefaradim recite Avinu Malchenu skipping the parts which mention sin (see Shut Yechaveh Daat 1:54. Piskei Teshuvot 550 note #37).
Q: When does the fast end?
A: After three medium size stars have come out on Shabbat. Rama, Orach Chaim 249:4.