@Neot
In your first (and second and third) screen shot:
You should not have Email fetching both enabled and Fetch on Auto-cron.
You also have your default MTA set to use PHP mail function.
In the fourth/fifth/seventh screen shot your hosting provider says that either you are using the wrong credentials or the account is not allowed to log in @localhost
In the sixth screen shot
What settings do you have on the Remote Mailbox tab?
Also uncheck header spoofing.