Sorry, I don't have an answer for you at this time, but I have had a similar issue with Gmail configured in the ASUS Email app. If I send an email, it seems to get stuck in the outbox. However, when I accessed my Gmail from the web interface, I noticed that the same email showed up in the Sent folder.
For me, the odd thing about the mail app and Gmail is that if I delete any message or send an email, it appears in my web Gmail account with a new 'Imap/trash' or 'Imap/sent' label, but the mails deleted via the app are not actually getting deleted.