mail

If your account is an IMAP account then the email on the server is not removed. Otherwise your email would not be able to synchronized across all devices accessing the account.

If your account is a POP account then the default behavior is to delete the email off the server when the first device accessing the account downloads the email to the device. There is a setting in the email client for a POP account to leave the email on the server even after download.

I think you have to confirm it is a POP account and that your client is not configured to leave email on server.

What is the exact Path to Trash, Mail>Preferences>Accounts>gMail>Mailbox Behaviors, what path is Trash Mailbox set to?

Ah, thanks, so long since I've had any POP accounts.

In a Browser log into your Webmail & check he destination of Trash, like make sure it didn't get dragged Archive or other Location.

Then delete in WebMail & see what happens.

The setting is to save the email on the server or not . I don't use Apple Mail but I have never seen an email client that has a setting to delete email from the server when quitting the email client. What email client are you using?

Once again, I don't use Mail as a client so can not be certain but I do know that the POP protocol downloads and by default deletes email from the server. I don't know what that setting is that you show. You should not have to make a choice to remove a POP message from the server after retrieving it. It should be the default behavior. In my email client if I was using POP protocol the setting I would make would be something like "Leave email on server after download." or "Do not delete email from server after download." It seems strange to me that you need to select the default behavior.

Sorry if I'm off on the wrong track. It just seems strange that you would have to select the default behavior and that by default the email would be saved on the server which is opposite to the expected protocol behavior.