Not rare at all. The UK convention for almost all dialects is to use cinema.
Depends on the dialect. Where I live, people usually call it a cinema and to a lesser degree a movie theater. In some areas, people also call it "the movies". Theater sounds a little strange to me because you could be talking about the place where you go to watch plays.
In Ireland the cinema is often referred to as "The pictures", it is never called the theater. The term theater refers to a drama or concert venue.
And here in Alaska, I usually hear (and always say) "the movies" unless the conversation is specifically about the building in which movies are shown - that is, I would say "The restaurant is just passed the movie theater" but "After dinner, we're going to the movies."
Just for what it's worth, I'm 38 and lived inTexas my whole life and I have never heard anyone say "I am going to the theater" refering to a movie theater. it's always been "going to the movies", " going to the movie theater" or "going to the cinema". If nothing else it goes to show how different coloquial terms can be, and I think DL is right to accept only the most universal forms.
Howevever, the term is not that colloquial either. The movie ads and the media often say "in theaters near you", "hit the theaters", etc.
I see a lot of comments about who says theater versus who says cinema and the like. I think the point that isn't being made clearly is that unless you are in the extremely rare circumstance in modern life where you are going to watch a play or an opera, when someone says the word theater, it's just known that the person saying theater means a place that projects movies on a screen to a room of people. In fact, the only times I have heard anyone say theater and NOT mean movie theater, they instinctively added "to watch a play" to make it understood what they meant. No one regularly says theater and means anything but movie theater unless it's part of their job, and/or it is common for them to not see movies. DL is trying to teach us the meaning of the Russian, but if that is to be successful, I feel they should put just as much effort toward the clarity of the English portion of the lessons as well.