I've only ever heard cinema refer to movies. But it's very rarely used at least in the part of the US where I live. People will usually just say they are "going to the movies," and we usually refer to the actual establishment as the "movie theater". I think where I live you would be looked at as odd for using the word "cinema."
So, theater is both, but you would call the one where you watch films "the movie theater" and you'd really not call the overall establishment just "the theater" (though an individual room might be called that, as in "Star Trek is in theater 5"). And cinema is rarely used at all.
Oh and to answer the OP, I pronounce it thee-uh-terr, but like someone else said, with a short "uh"