"I go to the movies every week."
Translation:Minä käyn joka viikko elokuvissa.
Joka + X in time expressions is always nominative: joka päivä, joka viikko, joka maanantai, joka toinen vuosi (every other year), joka viidestoista minuutti (every 15 minutes)...
I can think of some ways to use joka + something other than nominative, but in those cases we're really talking about a shorter way to say "jokainen", which behaves more like an adjective. In this particular case, "Käyn elokuvissa jokaisena viikkona" or something similar would sound strange. "Joka viikko" is a much more concise standard phrase.
It doesn't have to be. The word order in Finnish is quite free and you can express different meanings with it
Minä käyn joka viikko elokuvissa = "the movies" is the central new information: I go to the movies regularly and I'm going to tell you something about that experience. This feels the most natural way to translate the English sentence to me.
Minä käyn elokuvissa joka viikko = very close to the previous one, but the "every week" part has a bit more weight, makes me think that the conversation is going to address that rather than going to the movies in general. I'd argue it should still be accepted as a translation here.
Bonus: Elokuvissa minä käyn joka viikko = To the movies (as opposed to anything else), I go every week. This kind of nonstandard order can be used to emphasize certain words. It's already specific enough that I wouldn't say it needs to be included as an answer.