question about game center

Yes it removes all the Game center achievements. I recommend you don't remove them from gamecenter. The gamecenter stats barely take any space on the phone.

 

If you just select to remove the app, afterwards it asks you to delete from Gamecenter or not. If you do that it'll remove all your stats so don't do that unless you want to remove all your history.

 

If you change iphones all the achievements will be erased, unless you restore from an iTunes or iCloud backup. If you restore from a backup everything will be identical in the new phone.

If you remove a game and re-install, it will be dependent of the game. Gamecenter can communicate both ways, but most games only send data and do not read the stored data.