Translation:Do you want to watch a movie?
I do not know whether the original question was edited when you commented 2 days ago, but presently, it reads the same as when I replied 5 months ago, viz., "Is there any way besides context to tell this sentence apart from the more general question 'Do you like to watch movies?'" I believe fhwqhgads' reply, "Say 你喜欢看电影吗?" is, indeed, a suggested translation of "do you like to watch movies?," in reply to yarrite's question, rather than 你想看电影吗, the original sentence of this DuoLingo exercise.
"To go watch" is not necessarily incorrect, if that is really what you mean to ask. For example, if you and a friend are at home, and you ask if your friend wants "to go" to the theater "and watch" a movie, you could certainly ask, "Would you like to go watch a movie?" but even in that case, "go" is superfluous: you don't need "go" even in that sentence. However, if you and a friend are at home, and you want to show your friend a movie on your television set that is right there in the same room, then it would be wrong (misleading) to say "go watch" rather than simply "watch:" simply, "Do you want to watch a movie?" is better, in that case.
In the case of 你想看电影吗, there is no context; so, “Do you want to watch a movie?" is a better translation, as "go watch" is not in the original, even if the question were asked in a context in which "go" is implied.
I think both sentences could mean "Do you want to watch a movie?" but 你要看电影吗 could also mean "Are you going to watch a movie?" or "Do you intend to watch a movie?" Of course, I could be wrong about that. To ask, "Would you like to watch a movie?" or "Do you want to watch a movie?" I would say, 你想看电影吗, but maybe that's just personal preference. I think that question, the one with 想 is polite, and "safe" that way. As for "formal," to me, it is formal enough for anyone I would ask that question of, anyway, that is, to anyone I would actually ask to the movies, or ask to watch a movie with me. I suppose you could make it more formal using 您 rather than 你, or even more formal by substituting the person's name and title for 你, but I do not know that 要 is any less polite or less formal than 想.
"See" a movie is the same as "Watch" a movie. Are the ones writing this APP not very good speakers of English? There are way too many correct answers in English that the APP flags as incorrect. The point is the meaning in English and not "word for word" meaningless translations.
To be fair, there are many ways of saying things in English, (Do you want to see/watch a movie/film?) and it's difficult for them to cover all of them for every question. I think sometimes native speakers of English forget how rich the English language is in terms of vocabulary and ways of saying things. I'm generally very happy with this course. There is a report button to report issues such as this. People will point them out, and as they do so things will improve.