Kaifeng lodging--Others will have to jump in and help you here, as it's been some time since I've been there and I didn't stay in commercial accommodation.
Beijing polllution--nothing you can really do to prepare, and you may see some blue skies as well. Generally, the windier it is if coming from the north or northwest, the better the air quality.
Beijing weather--getting cool but usually nice, one of my favorite periods of the year. You'll need a jacket but not a heavy coat. You can definitely leave the shorts and sandals at home though. Night temperatures will be chilly though not usually freezing. As for GW tours to remote parts, I should think somebody will still be running them all the way through October. Who have you contacted so far?
