Hi,
the error "the requested name is valid but no data of the requested type was found" looks like an error message returned when trying to connect to Gmail server ... does it happen only when you just started or woke your PC?
About the deleted mails - where do you see them? In the Inbox or in the root search? Note that the root search may include emails from all Gmail folders, including the Trash. You can switch to Table view (if you are not already in it) and then make the column "Folder Name" or "Folder Path" visible and you should see in which folder those emails reside.
Jogy