If you buy your tires from one Tires Plus, can you goto any Tires Plus to get them rotated and balanced?

Yes

Yes you can once your in computer they can pull up your name at any of them.

Balancing is free. Just make sure they actually do it. I mark the tires to make sure they rotate. I also get the road hazard for repairs

Yes, rotation is free, balancing may be an additional cost, not sure.