If I'm controlling an opponent's turn, do I get to chose where their commander goes?
Yes, you do get to choose.


  714.5. While controlling another player, a player makes all choices and decisions the controlled player is allowed to make or is told to make by the rules or by any objects. This includes choices and decisions about what to play, and choices and decisions called for by spells and abilities.


Where the commander goes is a choice they're allowed to make by the rules. There's no special exception or anything here, so yup, you can let their commander go to their graveyard if it dies in combat.