What is/are the Torah, Rabbinic and halachic source(s) for the prohibition against going into a church? Does this prohibition apply to a Moslem mosque as well?
The primary source for the prohibition is the Gemara in Avodah Zarah (11b), which states that it is forbidden to enter a house of idolatry — a ruling that is mentioned by later poskim with particular stringency (see Shach, Yoreh De’ah 149:1; Chochmas Adam 84:16).
Some write that this is a full Torah prohibition (see Shvilei Daviv 154), and Shut Peri Ha-Aretz 2:4, whereas others state that it is a rabbinic prohibition.
The prohibition against entering a church relies on the assumption that a church is considered a “house of idolatry.” This assumption is made by the Rambam, and by later halachic authorities, though there is room to discuss the status of different churches that have come a long distance from the original Catholic church.
The prohibition does not apply to a mosque, which (almost) all concede is not a house of idolatry.