It's just hasn't been added yet. For that feature it would have to continuously poll in some interval in the a background for each account. Since the websocket API is removed.
Its a bug where it fails retrieving the account info, it makes it look like ur not logged in. It's more notable recently because instances are having more issues.
Strange I just tried reproducing this but i was unable to do so. Does it only happen in Jerboa? Maybe a powersaving feature of your device is causing it. I think I might have idea to fix this. You will have to make an issue on github with your specific device and anything relevant. And then I can send you some test builds to see which resolves it.
That button has never done anything, it's functionality has yet to be implemented. There is draft in working, when ever that gets finished it will be in Jerboa.
Make a issue with your crash logs on github. I can't reproduce this, so it is probably something device/ android version specific. So the logs are really important.
There is a button to mark as read, aswell a mark all as read button.