Why does running the App Store push email to my iPad?

At an educated guess, I would say the following is happening:

Exchange is designed to push - and the exchange server is no doubt pushing to Apple servers - regardless of whether you have push enabled on the iPad or not, it's just how it works.

When you connect to the Apple store, you connect to the Apple notification server (or renew your connection to it) and in doing so pick up any mail that hasn't been collected from the exchange server yet. I would imagine the same would happen with other push notifications as they pass to the Apple server whether they are enabled on your iPad or not.

What would be interesting to test is whether this happens with a conventionally configured Gmail account - which would probably shed more light on what is happening.

This sounds like a bug, have you restored the iPad?

There is no reason why launching the app store should affect mail. I would restore the device and then see if it continues.