Syncing email accounts
I use my Gmail account with Apple Mail on an iMac, two iPhones and an iPad. When I delete a message one device, it doesn't delete that message on the other Apple devices.
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There are two very different methods of email management, both of which are offered by gmail.
The POP3 method, which you’re using, stores messages on each of your devices accessing the email.
The IMAP method, described here:
https://en.wikipedia.org/wiki/Internet_Message_Access_Protocol
… stores messages on the access provider’s server, so they appear the same on every device you use.
Switch to IMAP, and you’ll be fine.