Together (вместе) is an adverb, and I think adverbs (almost) always go before the verb/adjective. Мы вместе смотрим фильм = we together watch film. In English we usually put adverbs at the beginning if they are long ("Unfortunately, I cannot help you") or at the end if they are short ("I run fast"), but in Russian, they normally go right before the verb.
"See a movie" is used more in terms of planning. We might say, "We are going to see a movie," "We are seeing that movie tonight," or "I don't want to see that movie." But we don't generally say, "We are seeing a movie together" when the context is one of being physically seated in front of a screen.
Am I right that because смотрим is imperfective this implies a habit or recurring event?