Actually the way I learned from Oxford Grammar is that if you go to see a play, you could say "go to theater", while "go to the theater" implies going to the building, but not necessarily for the purpose of seeing a play. Similarly go to (the) school, being in (the) hospital, without the definite article implies going there for their primary purpose, with the article emphasises going to the building or location, may be for some other purpose. I found that this is not a hard rule, especially in the US, but there is a tendency of using one over the other depending on the purpode. But either way, "go to theatre" is idiomatic at least in some English variations (Google search provides many examples).
Not quite, that would be 'gå på bio', not 'gå på teater'. The difference is that 'teater' refers to a theater in the more classical sense of the term (that is, somewhere you go to see a live performance of some sort such as a play, concert, or comedy routine), whereas 'biograf' is used to refer to what in English is sometimes called a 'movie theater' (as well as 'movie' in the sense that we typically use it in English, often shortened to 'bio' in vernacular speech and the specific fixed phrase 'gå på bio').