I know I haven't responded to this thread in a while. TLDR: Things still don't work.
The only reliable way to get Google to sync is to disable celluar data and auto-sync, restart and then re-enable these settings. I also turn off my phone at night and when I turn it on it usually syncs so Google is never too far behind.
I have basically accepted this. I know that that sounds stupid but it's just a phone and simply not worth the headache. I have tried a variety of things, even going as far as reflashing the firmware. It just won't keep syncing. It might work fine for a day or so and just like that it stops working.
Meanwhile, all other features seem to work just fine, including the synchronization of my Exchange account.
Just installed 1.8. Maybe that will fix it. Maybe it won't. I hope someone else figures it out, not so much for my sake as for theirs.