So, my question is, do I have to click 'edit' and then mark each one for delete or go down the list and trash them as they open?
Are you trying to do this from the All Inboxes screen? If so, try going back to Mailboxes and selecting one account at a time. I have only an iCloud and a Gmail account on my phone, and when I view the inbox for each individually, clicking Edit enables three options at the bottom of the screen: Mark All, Move All, and Trash All (or, in Gmail, Archive All).
