I really don't understand your questions but I'll try to answer them.
Emails ARE NOT deleted from the server by my phone's email client because that's the way I want it. My computer is not relevant, however they are not deleted from my server by my computer either. The server is set to delete emails after 15 days.
My email is a private company hosted at GoDaddy.
When I delete emails on my computer they don't get re-downloaded to the computer. That's the way this is supposed to work. The phone is supposed to work the same way but it does not. It used to, on the HTC Droid Incredible. However there is a bug in the mail client on the Samsung Note 3. It does not seem to be consistent. However on multiple occasions I have picked up he phone and found that it had pulled over 400 emails because of this bug. And then I have to delete them again.
No, emails do not come back to the phone if deleted from the server.