I setup Google Gmail using Google's way to 'Set Up Your Apple Device for Google Sync' and it works just fine without prompting for any passwords. Nice since I have an Android phone which no both Calendar and Contacts sync all the way around no matter where I change them.
As for Facebook push, I loaded the iPhone Facebook app which does badges, alerts, and sounds, but I don't use it for Facebook. I instead just open it to clear the alert and then use Safari or the MyFriends HD app to view and respond in Facebook.