Are barracks soldiers allowed to make a request to switch rooms, or get a new roommate?

Yes you can. Ensure that you have a good, valid reason to request this switch.

Absolutely, speak to your first line first and they will relay to your 1SG. Be aware that you’re gonna need a reason more than just “I don’t like the person” or “ I want to be with my friend”

You are allowed to request it. Sure. That's just a conversation. It's not guaranteed it'll happen but you miss every shot you don't take.