You're probably thinking of entering a Mormon Temple. The Church of Jesus Christ of Latter-Day Saints distinguishes between churches where people meet every Sunday, and Temples where more important ceremonies are conducted (and which, incidentally, aren't open on Sunday).

Yeah, technically all Christian churches do not allow non-Christians to take communion, though it's on the honor system. However, I don't think an agnostic or atheist would miss a cracker and shot of juice.

I guess maybe wedding might be a niche concern. Some churches only marry believers or even members, but I bet most would consider it anyway. Only some sort of premarital class or counseling is typically required.