Outlook

http://superuser.com/questions/586914/re-download-all-messages-from-gmail-into-outlook-2013

File > Account Settings > Email Tab > Select your IMAP account and double click it > Adjust "Mail to keep offline" to All

http://answers.microsoft.com/en-us/office/forum/office_2010-outlook/duplicate-emails-in-the-gmail-sent-folder/3f2c4f3d-d006-48bd-9a8f-199e519862a6