“Fire” a weapon before firearms existed?

"Loose/release/shoot an arrow" or "shoot a bow" are all possible variations that avoid the use of "fire".

an ngram search provides some clues to the usage. "shoot" seems to be the commonest usage however all the others are possible.

As for the movies using "fire!", I don't know about Frozen but at least in the other two the people involved would have been using a different language, not English, so any speech in the movies can be considered as a modern translation and, as long as you consider the use of "fire an arrow" acceptable in modern times, it should be acceptable in the "translated" speech of the movies.

You are likely right.

The words to use would be Nock and Stretch according to this one

http://walternelson.com/historia/2006/06/another_historical_pet_peevear.html

Hollywood loves to have groups of archers draw their bows to full span, and then have them hold their bows drawn while waiting many suspenseful moments for their leader to drop his sword to signal the deadly storm to follow. (Think "Lord of the Rings") Here's the problem with that:

In choosing a longbow, you should pick one that is hard to pull. Not back-breaking hard, but it needs to give you some real resistance. This only makes sense. A bow is only as strong as the archer, but if the archer selects a wimpy bow, then his strength is wasted.

Now, when you are pulling one of these hefty bows, pulling it pointing it and shooting it aren't too tough. What is tough is holding it at full span. Unless you are shooting a toy bow, your arms will almost immediately start to quiver and the pain will begin. Furthermore, your bow doesn't like staying at full draw any longer than it has to either, and you could do it some real damage.

The only documentable medieval words of command for archers seem to have been "Knock"[sic] and "Streach"[sic]. These make sense, and presuppose that telling an archer to draw his bow is essentially the same as telling him to shoot it.[Bold is mine]