Grocery/List app for iOS with decent web application
Take a look at Catch. It was recently one of the featured apps on the App Store. It's not designed for shopping per se, but it can do everything but #2. Instead of a push notification, it silently updates. It does do notifications, but those are based on timed events.
It supports spaces and hashtags for categorization, which I think would fit your needs.

Also, it has a feature to share lists between people. If Catch doesn't cut it, you might want to look at Wunderlist also. Different, but many of the same features.

And, no, I have no connection to them, just like the app.

Update/Edit: Catch is no longer available. Wunderlist is better than ever, and Shopit the app is currently very hot (it's on the top free apps list in the U.S.) which does everything you asked about except sharing.