Mail stops syncing after certain date

You cannot set up Mail to “sync” with a POP mail server. POP mail clients can only download messages from the server and remove the messages from the server afterwards. Whatever you do to the messages in the mail client (i.e. in Mail), you’re doing it to the downloaded messages, not to the messages that may still remain on the server.

In general, what happens to the messages on the server after having downloaded them depends on your Mail > Preferences > Accounts > Advanced > Remove copy from server settings, although in the particular case of Gmail, those settings have absolutely no effect on the mail stored on the server. Gmail ignores any requests to delete messages from the server made by a POP client. What happens to the messages on the server after Mail has downloaded them is determined by the corresponding Gmail’s setting alone (Gmail’s Settings > Forwarding and POP page). The only way to manually delete a message from Gmail’s server is logging into your account on the web with a web browser.

Also, keep in mind that Gmail doesn’t allow the same message to be downloaded more than once via POP.

2. i've been deleting emails from the mailbox but i've notice that
everything i open mail or perform the synchronize account function
some deleted mails reappear in the mailbox.

What you say here doesn’t make sense to me. For starters, there is no “synchronize account function” for POP accounts in Mail.