Not sure.... my account is (as you will have seen) a Hotmail account, and as far as I'm aware nothing to do with g mail
On the home screen is an Email icon, and they come in via that, in my Hotmail account. (They arrive at the same time on my PC and tablet...)
I don't understand what you mean by: Swipe left/right to All tab next to Downloaded, Installed etc tabs.
However, in Application Manager I can see all the icons that are on the Home screen - except that the one I mentioned above - white envelope with @ sign, labelled Email - is not in the list!
When I tap any of the others I can see the Force Stop - but Email is not there to tap.
However, I have just Forced stop and cleared the cache from Gmail, and restarted the phone - and the problem has disappeared.