The message suggests that Thunderbird is trying to write to folders that don't exist. That could explain why your messages don't get saved in the appropriate folders: these folders can't be found.

I notice it is a .edu account. Is there not an IT or support department that can help you?

I do in fact have a trash folder but when I press the "delete" button it goes up in a cloud of electronic smoke, and my Trash folder is always empty. I mistakenly deleted an important message because I tried to delete the message above and Thunderbird didn't respond. So I hit "delete" again and it deleted the intended one, and the one blow it a fraction of a second afterwards. I tried to retrieve the latter one but it was gone.