so it seems you are getting the mail you are sending, but you don't see your old, or, "archived" email messages. is this what you sat saying?
is this is the case, you need to tell the app how far back to sync.
from the gmail app,
settings, then choose the account you want to edit,
scroll down, make sure "sync gmail" is ticked, and scroll to "manage labels".
go in there and make sure the folders or "labels" you want to sync are set to sync.
there is also a "days of mail to sync" option also.
if you want to get notified, there is a "notifications" box you check. this will allow you to set up how you want to be notified (vibrate, led lamp, etc.)