Hello tuyosi, I have a hard time reading your English so maybe I'm not following you well.
Are you trying to use Gmail through your personal domain as a mask? Does Thunderbird connect to your Dovecot (IMAP/POP3) or to your Gmail account? If you are trying to mask your Gmail, you don't need Dovecot and the configuration is far simpler. On the other hand, if you really do want your own personal IMAP/POP3 server using Dovecot, and the problem it's just that you can't send/receive email from addresses from @gmail.com, maybe your rental's IP address is just blacklisted because someone was using it to spam. Please, use the following or similar test: https://mxtoolbox.com/diagnostic.aspx If you're blacklisted, you should contact your rental server support and ask them to do the procedures to remove it from the blacklist. Sorry if I'm getting this all wrong.
