Re: "Mail repeatedly downloads deleted Gmail messages"
- if you have recently set-up iCloud to sync Mail ... there may be a setting there you can change ?
- Could GMail be set up to keep your Mac Mail and GMail Inbox in sync with same messages in both places?
- You may need to log-in to your account at GMail and check what actual settings are currently in place there: Try: https://accounts.google.com/ ( GMail may have added new features ?)
- In GMail: At the top right, click Settings
(ie: the gear icon) > See all settings
Eg: If you are able to set that an email deletes on GMail when you delete an eMail on Mac ... then GMail will no longer have that message to download.
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iCloud User Guide - Apple Support
Mail User Guide for Mac - Apple Support
Apple Mail Support website: (info. / support / Conyact options tc.)Mail - Official Apple Support