Need suggestion - Autumn leaves

We are traveling in a group of 4 people during Oct 3-11 (9 days). Our plan is to see autumn coloured leaves, natural sceneries, and outdoor onsen (rotemburo) in Hokkaido (Sapporo, Daisetsuzan, Akan Kushiro), Yamagata (Zao Onsen), Oirase, and Bandai Azuma skyline. Our flight will arrive Narita, Tokyo on Oct 3rd morning.
We are having trouble deciding how we plan our itinerary. Could anyone advise me :
1) Where should we go first? Zao or Hokkaido?
2) In Hokkaido, should we travel by car or by public transportation?
3) Please recommend the must-see places in Hokkaido,.
4) The best outdoor onsen

Thank you in advance

I believe early October is a little bit too early for autumn leaves in Zao and Bandai. It is certainly too early for Oirase. The nearby Hachimantai area might be better. In Hokkaido, however, you should be able to see the autumn colors. For Daisetsuzan it is a little bit late, but there should be many other areas for seeing the colors.

1) Where should we go first? Zao or Hokkaido?

I recommend to go to Hokkaido first and then go to the Tohoku Region. If colors are early this year, you might be able to see them there, too.

In Hokkaido, should we travel by car or by public transportation?

For travel in national parks and in the countryside, I recommend a rental car.

Please recommend the must-see places in Hokkaido,.

The following page introduces the highlights in combination with ratings. Because of your limited time, don't expect to see more than two or three areas:
http://www.japan-guide.com/list/e1101.html

4) The best outdoor onsen

One of the nicest (bus sometimes crowded) outdoor bath in northern Japan is Tsurunoyu in Nyuto Onsen:
http://www.japan-guide.com/e/e3650.html