Okay an update, and possibly a different question.
I've just tried turning off sandbox mode on my test GC account, and a badge has magically appeared on the app icon to notify an active turn (that's another issue I forgot to mention in the original message: no badges...).
Now, I know I shouldn't use sandbox mode anymore, but on ios 8 and 7 I get the error "The requested operation could not be completed because this application is not recognized by Game Center" when attempting to authenticate with a live GC account. So, the badge is a good sign but I still cannot test the app properly.
Is there anyhing I'm doing wrong?
UPDATE 2: alright, I give up.
I created a brand new app id, tried it in all my test projects and it has exactly the same problems I mentioned in my first post: server communication errors, no notifications, etc.
Anything I change in Xcode or itunes connect doesn't make any difference: one bundle id works perfectly, all the other ones sort of work but with the same issues.
The only thing I can do for now is test my app update with the "golden" bundle id, hoping to somehow sort this out before release.
Any suggestion would be really, really appreciated.