I understand how frustrating this must be, especially after months of dealing with the problem and contacting Apple Support without success.
Because every incoming email is immediately followed by a delivery failure notification, the priority is to establish what is generating these messages. The notification may relate to a separate outgoing email or an automated mail process rather than the incoming message itself.
First, test iCloud Mail in your browser. Visit iCloud.com/mail and sign in.
Ask someone to send a new email from your Gmail account to your iCloud address. Wait for it to arrive, then check whether a delivery failure notification follows. If necessary, repeat the test and record the results. If the behaviour occurs on the web as well as in the Mac Mail app, that is useful evidence for Apple Support, but it does not by itself prove that the iCloud server is responsible.
Open one notification and examine the failed recipient address and any details of the original message. Compare these with your test email to help determine whether the notification concerns that incoming message or a separate delivery attempt. If possible, view the full message headers using Apple's instructions: View all email headers in iCloud Mail. These may provide clues about the originating mail server and the specific error.
Although you have already checked forwarding, you can review your iCloud Mail rules for anything unfamiliar using Apple's guide: Set up rules to filter email. However, given the steps you have already tried, I would not keep repeating basic troubleshooting.
The most important next step is to request escalation to a senior iCloud Mail specialist through Apple Support. Explain that the problem has persisted for months despite previous support interventions. Ask Apple to investigate the source of the repeated notifications, including relevant server-side mail activity and account configuration. Provide a sample notification, its full headers, and the browser-test results, with personal information redacted. Apple should investigate the underlying cause rather than simply repeat steps that have already failed.
To manage storage in the meantime, you can move the notifications to a separate mailbox to keep your inbox organized. This will not reduce storage usage. Deleting unwanted notifications and emptying the relevant Trash or Deleted Items folder may free some iCloud storage, depending on where the messages are stored and how they are synchronized. See Apple's guide: Manage your iCloud storage.
If you notice unfamiliar recipient addresses or other signs of unauthorized activity, review your Apple Account security as a precaution. Otherwise, focus on getting Apple to investigate the source of these notifications.
My recommendation: Gather one representative notification, its headers, and the results of the browser test, then request a senior-level investigation from Apple. You should not have to keep troubleshooting indefinitely when the problem has persisted for months.
