You can't recieve GMail and you're not getting an error message? Right click on the GMail account under the inbox and see if it has a selection to take the account online or offline?
Is Gmail setup as POP or IMAP?
If IMAP, expand the gmail account at the bottom of the sidebar and see what mailboxes are there.
It does have the online/offline option. It's online. I've switched it on and off and on (multiple times in the past, trying to get it started again) and recreated a duplicate mailbox (which only exhibits the same behavior... able to send email but not able to download received emails).
Gmail is set up as IMAP.
The mailboxes at the bottom of the sidebar are...
Thanks for your suggestions and pondering!
Notice that your IMAP mailboxes for Sent Mail and Trash are still in the IMAP folder for that account. That means those folders are not syncing with the website or any other device you're using with this account. Inbox is automatically set up with IMAP. All of the other mailboxes under Mail's main mailboxes for that account are only local mailboxes.
If that's how you want it, that fine. If you want it to sync between all devices and website, you will need to map the IMAP mailboxes to Mail's main mailboxes.
Ex: Select and highlight Sent Mail under IMAP folder, go to Mailboxes in the menu and choose Use This Mailbox For - Sent. The IMAP Sent Mail mailbox will be moved to Mail's Sent mailbox.