Okay, I've done some experimenting and narrowed this down some. The problem lies at
www.me.com. Over the last couple of hours, I've been reading my email on both my iPhone as well as on my WM based Moto Q. I've been able to read the messages fine and delete them fine on my iPhone (and Q). But for some reason they still appear in the .Me inbox, even though they are deleted. When you go into the inbox and try to open the deleted message, it will disappear and all other messages that you've deleted will disappear also. I would think that since the problem is on the .Me side, that Apple should be able to fix it easier than say if it was on the iPhone side.
Also, FWIW push calendar is now working fine between my iPhone and .Me. Push email still doesn't deliver though. I still have to do a manual fetch, whenever I want to read my email. But this is not a biggie to me.
I won't jump ship on .Me yet. I really would like to get this to work and hopefully the Apple folks are busy trying to figure it out. I really have a secret agenda here. By moving my email, calendar, contacts, etc. to .Me, that is my first step to eventually buying me a Mac (probably a mini, if they are still being sold).
PS - Sorry for repeating what island posted. I should have read more carefully. Anyway, this confirms it is a repeatable problem. That is the easiest kind to fix. I just hope someone has reported it to Apple (surely someone has).