How to transfer mails from Mac to iPhone / iPad , even they are signed in with same Apple ID and password say example1@icloud.com
In iPhone go to settings > in password and accounts > click on add account you can select the desired account it could be gmail , enter gmail address and password and sign in it will prompt to check calendar , mail , notes , slide the box of it the apps data will start syncing .
There is an option fetch new data - push > in iPhone
You can connect the iPhone and Mac using usb cable and sync them Sync content between your Mac and iPhone or iPad over Wi-Fi - Apple Support
Sync content between your Mac and iPhone or iPad - Apple Support (IN)
The mails from Mac / iPhone will sync and take time ( the fastest way is sync complete data ) they will be mirrored , you delete a mail from Mac it deletes in iPhone , make sure all the mails should be read ( no blue dots on each mail message ) Mark emails as read or unread in Mail on Mac - Apple Support
See the articles to delete mails from Mac Delete emails in Mail on Mac - Apple Support
And from iPhone Delete emails on your iPhone or iPad - Apple Support
You mentioned - If I delete from the devices the mail stays on the server and does NOT delete from my Mac, making it frustrating and more for me to constantly clean up.
The reason is your primary Apple ID is example1@icloud.com was created from Mac , and your gmail address was configured from it , you are not using primary Apple ID say example2@gmail.com , in case if it was used the syncing will not occur .
The mails will be deleted only from Mac , the iPhone is not a primary device it only syncs and acts as a secondary device .
Just to explain , there are few iPhones , and a single Mac that are synced on the same network in an office, the boss has created Apple ID from Mac A ( its a first device and Apple ID is always created from hardware address of it ) , the other persons on iPhone can only view the synced mails but can also delete from their own device it will be deleted , but not deleted from Mac and gmail server .
The credentials for gmail account / iCloud account are confined to the boss who is the owner of account ,and cannot use his account and make their own see https://support.google.com/accounts/answer/46526?hl=en
The odd part is on a 2nd gmail account I use for personal emails, the IMAP functions flawlessly across all 3 devices just fine.
These are more gmail account say the mail address is example3@gmail.com and example4@gmail.com they can be added in Mac system preferences > internet accounts .
See the articles Use your Internet accounts on Mac - Apple SupportUse your Internet accounts on Mac – Apple Support (AU)
Change Internet Accounts preferences on Mac - Apple Supportsupport.apple.com › en-in › guide › mac-help › mac
Add an internet account - Apple Supportsupport.apple.com › guide › mac-help › mac
