Can I receive local or push notifications to my email address? Can I receive local and/or push notifications to my email account instead of as a notification that appears on my iPhone? 

I want to receive the local and push notifications for some apps but I don't want to necessarily receive them as pop-up notifications on my iPhone screen, but would rather access them in my inbox when I have time.
The short answer is No.

iOS does not have any way for you to redirect notifications.

It may be possible to change how some notifications are sent, and have them sent to you via Boxcar or something similar, but that is going to be limited to a very small subset of applications or services which are designed to offer alternatives. Twitter and Facebook come to mind.