Is it required to just begin or to finish Mincha by Shkiya (sunset)? Does the same apply for Chazaras Hashatz? What should the Minyan do if the Chazaras Hashatz will surely go past Shkiya?

It is usually assume that one should complete both Mincha and the repetition of Shmoneh Esrei before the end of Zman Mincha, and if one assumes that Zman Mincha ends at Shkiyah, then Chazarat Hashas should be completed by Shkiya. If time is short, the “Chazzan” should begin his Shmoneh Esrei aloud and recite, with Kedusha, through Ha’keil Hakadosh, and then complete the Shmoneh Esrei silently. This procedure is often called “Heicheh Kedusha”- literally “Kedusha aloud”- meaning that the quiet Shmoneh Esrei is recited aloud until after Kedusha. There is a well-established rumor (perhaps totally anecdotal, perhaps true) that the Baal Shem Tov held that one is not required to complete Shmoneh Esrei by the end of the Zman, but it is sufficient to begin it by then. As I mentioned before, this is not the Halachic assumption of most authorities.