Mac Mail with gmail help (deleted items)

I use Mail on my iMac at work, but with gmail which is our corporate webmail. Here's my issue: If you look in the left-hand column on Mail, you will see the following trash or deleted items files:
Under "mailboxes": Trash (on my Mac)
Trash (iCloud)
Under "gmail": (gmail) Trash
Trash
Deleted messages
Under "iCloud": Deleted messages
Deleted messages (name@me.com)
Trash

I did not set these up, but there they are. Which ones can I delete, and is there a way to make sure they don't show up again?

And finally - the most important part - when I delete a message, it disappears, and does not show up in any of these! Where does it go?