How to transfer mails from Mac to iPhone , 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
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 mail from the devices the mail stays on the server and does NOT delete from my Mac .
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 is an iPhone and a single Mac that are synced on the same network in an office, the boss has created Apple ID from Mac A ( it's 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
