I would like to ask if there is any impact when you rename a group, do I have to keep the name that I started with when it is already in use? or I can rename any time I want and CommCare will update all the places the name is linked to?
CommCare typically references groups by their ID, which doesn’t change, so that should be fine. If you saved a group name to a case, that wouldn’t be updated automatically, though that’d be unusual to do in the first place.
If you have some particular concerns in mind, you could always try it out to double-check.