The reason I had you verify the mailbox behaviors setting on your Mac is to double-check that each of your Gmail "mailboxes" are corresponding to the mailboxes that you will see in the Apple Mail app.
As far as syncing between Apple devices, there are two things to keep in mind:
- All of these devices are logged in with the same Apple ID.
- All syncing, including email messages, is done between these devices via iCloud.
I'm also assuming that your Gmail account is configured for IMAP and not POP3 for accessing the Google mail servers ... and finally, that you have your Gmail account properly configured.
If all of these are true, then I'm afraid, that I'm at a loss on what the culprit is that not allowing your Gmail messages to sync properly. Maybe, someone here, who is more familiar with Gmail's functionality will chime in to assist you. Sorry, that I couldn't help. Good luck!