Those are a surprise, I hadn't noticed anything like that with Modern but Classic is doing that since we made the cookie change. On classic I figured out that you need to do an intentional logout with the "Login Status" link, then post to get a all new cookie. Then you stay fully logged in.
I don't think it could be the same on Modern though, they work too differently. I'm not seeing it with my username on Modern though I did intentionally log out and log in.
I'm going to have to think about the email issue, it shouldn't be able to do that.