Can I use Google Sync (Exchange, Push) through Mail.app in Mountain Lion?
No, you can't use Google Sync through Mail.app in Mountain Lion.

Google's exchange server m.google.com is not a fully implemented exchange server. It just implements the ActiveSync protocol which seems to be a subset of the exchange protocol for mobile devices. See also this information on Google Sync: http://support.google.com/a/bin/answer.py?hl=en&amp;answer=135937