I am playing tomorrow and am thinking of using a giant character. My question is whether giants can go inside. I don't want my big guy to have to wait outside if we are playing on the Lab map. Or do I? Thanks for any help and/or advice.

Nothing prevents the giants from moving inside. Feel free to check the Icons rule book section on Giants to verify this (pages 30 and 31). You probably should peruse it, anyway... Giant rules can be a little tricky and if you've never really used one before (and don't see them very often at your venue) it couldn't hurt to brush up on their game mechanics.

I read the Rules, FAQ and Errata concerning giants.

Am I to understand that all opposing characters up to two squares away from a giant character are vunerable to a Quake attack?

Would this reasoning also apply to Shake Off as well?

Yes, and yes.