Is it possible to have a separate password for Game Center - one that is not the same as I use for my Apple ID?
Yes and no :-)
- If you use your Apple ID to log into Gamecenter, you must use the same password as for all the other places you use your Apple ID to log in.
- You can create a new Apple ID on icloud.com though and use that one for Gamecenter
There is a way to do this with a secondary Apple Id and it is not very convenient.
- Logout > Sign in > Logout whenever you want to play or exit game center