This problem with the popup and not receiving new email really bothered me. But I found the solution to the problem. The solution was in lowering the security setting for Gmail. Even though Outlook and Gmail think this is a bad idea, the instructional paper I read said that Outlook 2013 does not support the tight security setting for Gmail. So I turned it off. After I did this, the popup disappeared and I got a backlog of 25 emails. I can breathe again.