What phone are you using? I use a Samsung Galaxy S I9003, and when I set up account sync, it starts a background service for syncing my gmail and google apps accounts, and is syncs and notifies just fine regardless of whether I have the mail app running in background(I generally don't since notification works without leaving it running).

Either you are using an older software, or you haven't enabled sync.

It sync since 1.0; three years ago I used it with G1 and it worked fine. There is no older version, that was made public.

GP has either disabled sync, or is trolling.

I am using a HTC ChaCha. Before this I was using the Milestone. It is a huge problem.

To the down-voters, pay a visit to the App Store and see the reviews.

it does sync; check your settings and check how much internal storage you have