Does the blotted out email in the last screen shot match the email in the second screen shot that you are using as the default MTA address?
ie is the the gmail address that you are using?
The email error says failed to set sender. This could be caused by using an email that gmail/google doesn't accept but that would usually result in a unable to relay.
This could be because the PEAR mail class in osTicket looks to be pretty old.
I dont know if this will work but If your feeling adventurous you can download a newer version from here:
https://pear.php.net/package/Mail/download
And try replacing the /include/pear/mail.php with the newer version and see if it fixes the issue.
I've also asked the devs to take a look at this thread.