If you go to settings in GoogleMail on the web, and go into 'Forwarding and POP/IMAP' you will see the POP and IMAP settings. Make sure that POP is disabled and that IMAP is enabled for receiving mails. That should sort out your problem. All received mail should be the same on the server, your iPad, computer etc. Deleting a mail will delete them from everywhere, remember.
However, I am confused by how GMail works for sending emails. It just seems to work like a POP account with an SMTP server, as all 'sent' mails just seem to get stored on the device that they were sent from. No syncing at all.
On my MobileMe IMAP account, all mails - received, sent, etc. - are all synced to all devices.