Earlier quoted context omitted.
Google definitely forbids self-updating apps on Google Play. But I'm not sure how well this is enforced.
Horribly. I get a few games from the Japanese market, and almost each one requires in immediate internal download and update. Although those updates never trigger the Android update service, so I'm not sure if they are just downloading more resources of if they are able to request new permissions(I would like to assume not.)
Google did recently change the Android permission model; previously, apps had to request all their permissions at install time and it was all-or-nothing (and frankly, hardly anyone bothered to look them over.)
Now, certain permissions have to be requested when they're needed (at least for recent versions of the SDK) and the user can choose to allow or deny. But an app can't grant itself new permissions without going through the official update process.