BR,
Welcome to the Discussions.
A Gmail account is a bit odd in its behavior. It is primarily designed to be used with web access, but does allow POP access. Furthermore, if you use the smtp.gmail.com as the Outgoing server, then messages you send from Mail, or any other email client, will also be saved in the Sent folder on the server, but this is not like normal behavior for a POP account, and more like an IMAP account. But, unlike an IMAP account, mail sent from other than your email client will NEVER be downloaded to the account on your computer. You can't even copy yourself, because Gmail will NOT let you send an email to yourself for POP access, even though it will appear in Inbox on the Gmail server.
No ordinary POP account will sync the Sent messages, either.
More questions, as needed, please.
Ernie