You're supposed to have your stick on the inside, if you're actually in the competition, not warm up then no, don't bother, concentrate on what you're doing and keep your stick in the hand you are most likely to need it
For me it depends on what horse I’m riding and how they’re acting.
You should when you're training, you shouldn't during competitions
You don’t have to let go of the reins just take it in the hand you have and like drawing a sword pull it to the side you need it to be on
No never change it because you have to keep both hands on the reins at all times