Sorry, no Google+ apps from me, for at least a month (have to be 18+ to join G+, which kinda makes it a fail imo).

Chat is obviously real time.
Noticed the no sound notification part, working on it.
You should still get email style notifications, check settings to see if GUI notifications are on, they're working fine for me.

Friend accepts/ignores were actually not tested, just implemented according to the API.

I have no idea how to do push notifications (doesn't it need a remote server?)