I have the same problem which is why I stopped in this forum today. I've only had my S4 for a couple weeks and I'm constantly deleting my emails, but they just won't go away. I thought it had something to do with the fact that I also have a Samsung tablet and hadn't deleted the emails from it but maybe that's not the case.
I'm not even sure I can delete emails twice and actually get rid of them. I could have sworn some of the emails showing up today got deleted a few times but I never really thought about it until this morning when after clearing out all my emails last night, I noticed they were all back today.