For sending, since you were seeing it with another app as well, it sounds like a server issue.
Since it's Samsung's Email, you can contact BT's support and they won't tell you to just go away (and they almost certainly will if you mention a third party mail app).
On the second issue with Deleted -- the way it's supposed to work, Aqua syncs with the mail server, Outlook (and whatever else you're using for this account) also syncs with same mail server, and Aqua and Outlook may *appear* to sync with one another, but they really don't.
Please use web mail to determine which side of this scheme still works (Aqua <-> web mail, or web mail <-> Outlook) and which is broken.