Unable to send with Gmail account

First of all, MailMate does lack some additional visual feedback with
regard to the state of outgoing messages. But sending should work for
Gmail -- and if it fails to send then a window should be opened which
states the problem.

I'm not sure what the 'Outbox' folder is. Maybe the 'Outbound' mailbox
in MailMate? This is a virtual mailbox which simply shows messages which
are scheduled for sending. The messages are still in the Drafts mailbox.
When sent they are moved to the assigned mailbox for sent messages on
the related IMAP account (for Gmail that should be 'Sent Mail'). Gmail
is not a typical IMAP server which you can read more about in this blog
post.

You should try opening the Activity Viewer (⌘0) before sending a
message and then see if the log of the connection to the SMTP server
contains something useful.

You can edit/see the settings for the account (including SMTP) by
opening the context sensitive menu over the account and select 'Edit
Source…'.

You can see/change the mailbox type (drafts, sent, archive, junk, ...)
in the context sensitive menu of each mailbox of the IMAP account.