Dude, this is f___ stupid. I got it to work but it took a lot of trying. Go to Settings -> Mail -> Accounts -> pick the email account and then remove the green slider for the Mail icon. In my case it hung up for about 5-10 seconds. (Don't force kill the app. Just wait. This seems to be a norm for Apple software these days.) Then when it comes back to life, turn Mail back on. In my case when I then loaded up the Mail app all deleted/stuck messages were gone.
It seems like it started happening after this last update to v.10.2.1. It also feels like they break more stuff in those updates than they fix. So it will make me think twice before I click "go ahead, stop annoying me and install your update" button next time...
Anyway, it's time to ditch Apple Mail app. Anyone, suggestions for a better alternative? (I need it for gmail and hotmail accounts, and one icloud.)