Sent emails don't appear in Sent folder
Whenever you encounter a problem with Mail.app, the first step is to rebuild culprit mailboxes.


  When you rebuild mailboxes for IMAP or Exchange accounts, messages and attachments stored on your computer are discarded, and then downloaded again from the mail server to your Mac. Your mailbox appears empty until the download is complete.


It is a similar fix to clearing your browser cache when you encounter website problems.