Yes on both. I assume the settings are right, but I still just get an e-mail once in awhile on my iPhone.
I would highly recommend a Gmail account. Free, tons of storage and very reliable....not to mention some of the best spam filtering you will find.
Trick is to always have a "trash" email address that you use for forms and whatnot and those people that like to fwd you and send fwds with you on the list.
Then have one or two main emails that you use on your phone.
You will have to turn off recent mode right now to not get send mail notifications with Gmail.
I cannot wait til' mail is done right on the iPhone as I am used to a Blackberry.