Mac Mail imap syncing with Gmail's sent, drafts and trash can be quirkly, especially when you mix in an iPad.
First, be sure to go into the 'manage labels' in you gmail online settings. CLean up any folders, specically, the 'sent messages' folder that Mac Mail can create. Also be sure the the real gmail 'sent' folder is 'shown in the IMAP list' It's a checkbox. Then, in Mac Mail, select folder you are trying to sync out of the gmail IMAP list in Mac Mail. Then, while it's selected, go to the 'Mailbox' Menu, choose 'Use this Mailbox for" - select 'sent'. The folder will then move up to the top left Mailboxes list and be uses across Mac Mail and Gmail.
Kurt Rein Mobile Computer Wizard 619 255-1215 Office Computer Repair San Diego