Hi just4Lance,
Thanks for using the Apple Support Communities!
From your post, I understand that you have some accounts in Mail that are not behaving as you would like. It sounds like these accounts may be set up as POP accounts. If so, the behavior you describe would be expected.
The way that POP accounts work is that everything is kept on the server until it is downloaded to your computer or device. If you have more than one device, like you do, then you will need to delete them from all devices.
The other type of account is an IMAP account. The way this type of account works is that it will remain in sync across all of your devices, providing the devices are connected to the internet. The emails are always stored on the server and when you check your email, it will verify that everything is the same. So if you delete a message on your iPhone, it will also be deleted from the other devices.
Some accounts are configured to use either POP or IMAP and don't offer a choice of what kind of service you would like. I suggest contacting the email providers for information if it is possible to set these accounts up as IMAP accounts and how to do so if it is possible.
For information on configuring your Gmail account with IMAP, see this Google Help resource--Set up Gmail with Outlook, Apple Mail, or other mail clients - Gmail Help
This information above and more can be found in the following link.
Mail settings you might need from your email provider - Apple Support
Best!
