Are you allowed to vote inside a church? (Answers supplying sources would be preferred, as always.)
According to this answer at the Institute for Dayanim (an organization I had not previously heard of), voting in a church, not in the sanctuary, is permitted if that's your only option, but you should try to avoid it if you can. They don't address how far you need to go to avoid it (e.g. do you have to leave town for the day so you can use an absentee ballot instead?).
With regard to entering the basement of a Church for purposes of voting, see Teshuvos V’hanhugos 2:410 and Chai Ha’Levi 4:63, who both write that one can enter the basement for purposes of voting, providing there is nowhere else to vote. However, the case of voting is special, because it is known to all that the place is used for voting, as the poskim mention.
Note that many poskim write with great stringency about entering a Church, and Rashba expresses doubt even concerning entering in cases of pikuach nefesh (under such circumstances, however, Rosh explicitly permits entering). For a leading Sefardi ruling, see (at length) Yabia Omer, vol, 2, Yoreh De’ah no. 11. See also: Birchei Yosef 149:2, Darchei Teshuva 150:2, Igros Moshe 3:129, Minchas Elazar 1:53-3, Yechaveh Da’as 4:45, Binyan Tziyon 1:63, Shearim Metzuyanim B’Halacha 167:9, Rivevos Ephraim 3:496-497.
On distancing oneself from a church in general, they cite: Rema Yoreh De’ah 149:2; Iggros Moshe 3:129; Shulchan Aruch Yoreh De’ah 150 and Darchei Teshuva 150:2; Chai Ha’Levi 4:63.
As long as you don't go into the main sanctuary there should not be a problem.
There is a makhloket poskim as to whether belief in the trinity is forbidden only for a Jew. Many say a non-Jew may follow these tenets since belief in the omnipotent G-d is still present.
However, it is forbidden for a Jew to enter the sanctuary of the church, i.e. where the actual prayer services are held. As it is a marit ayyin as it could be interpreted as identification with the philosophy. However, it is permitted to enter other rooms in a church for non-religious purposes.