Cinema vs Movie Theater go with Movie Theater. 

However, we usually just say movies. Like "Do you want to go to the movies?" or "I went to the movies."

I guess it might be used to say "I work at a movie theater", but most other times just say "the movies".

If I'm not mistaken, this is how an American English speaker might phrase the following sentences:

1. Let's go to the cinema (Br.E.) = Let's go to the movies (A.E.)

2. My office is opposite the cinema (Br.E.)  =  My office is opposite the movie theater (A.E.)

3. The history of celluloid is inextricably linked with the history of cinema   ( Both Br.E and A.E.)

In other words,  A.E. uses 'go to the movies' to refer to the activity, a 'movie theater' to refer to a building, and 'cinema' ( in formal contexts) to refer to the genre/art form.  Please correct me if I'm wrong.

Yes, we do say normally just "go to the movies" if we mean we are just going to see a film, without specifying where. The "movie theater" is the building you are going to and could be anywhere, or there could be several to pick from.  So, if you said, "Sure, let's go to the movies" you could also add, "And what (movie) theater are we going to?" This is because you could have 5 to pick from. Or, you might ask someone, "Hey, where's the movie theater?" 

It's true we don't say we're going to the "cinema"; however, many movie theaters call themselves for commercial purposes the-- "XYZ Cinemas" for example, almost always in the plural.  That's very common to see. Some just call themselves the "XYZ Theaters" (again, usually in the plural). The word "Cinema" to refer to the actual name of the movie theater or company running the business is very common in the US. So,  the word is not unheard of in the US at all, it's just used slightly differently.