How is it possible to send a message and have it NOT be saved in the Sent mailbox folder when using RoundCube webmail?
When composing a message, select the drop-down box on the right next to "Save message in". By default it says "Sent" but you can click it and select "- don't save -" or another mailbox.
Note: this does NOT affect any logging. The message is still logged normally, and the absence of a message in the Sent mail should not be an indication that the account did not send the mail. The log at /var/log/exim_mainlog should be the authoritative source on what account sent a message.