Unfortunately, I do not have an answer for you. I'm sorry. .. BUT, I figured maybe if I posted and complained about my issues with syncing, emails, and stupid Google admin rights that forced me to not be able to access my work e-mail anymore (SUCH AN INCONVENIENCE:-[ ) maybe we can get some responses!!!
I orig only had the icon with the G in my notifications but I was still receiving e-mails. Then my hangouts stopped working, which is the absolute worst thing that could have possibly happened to me bc we use it at work a lot, and if I'm not at my desk I could answer quickly on the fly. I'M SO ANGRY ABOUT THIS. btw I have an HTC One M8, S-OFF, running on official Marshmallow. I tried to uninstall, disable this stupid "Google Admin Rights" app, it worked for a minute once I took device administrators for Google out of my system settings but then went right back to NOT working. Then I tried removing my account and readding it back in, bad idea... IT'S GONE!!! I currently have my email set up on my work phone, which is an iPhone that I don't use much and I can't use hangouts on there either
I know there is a way around this I just haven't been able to figure it out.
ANYONE WHO HAS ANY INSIGHT PLZZZZZ HELP? I'M ON MY HANDS AND KNEES LITERALLY