Unable to delete emails and sent messages disappearing in Apple Mail

1. I can't delete mail from my inbox.
2. Mail that I send seems to go out okay, but when I look at my sent box, the mail appears momentarily and then disappears.
3. When I am creating an email, every 30 seconds or so, this message appears on my screen, blocking me from doing any further writing until I hit "return" key

any help will be appreciated.

[Edited by Moderator]

Original Title: problems with Apple Mail