Hokkaido Rail Pass

I'm aware that Hokkaido Rail Pass covered all line except the Shinkansen. However, I would still like to confirm if the following trips are covered by Hokkaido Rail Pass:
1. New Chitose Airport to Hakodate
2. Hakodate to Sapporo
3. Sapporo to Asahikawa
4. Asahikawa to Sapporo
5. Sapporo to New Chitose Airport

Also, will it be cheaper to get a 7 day pas or to purchase separate train ticket.

The 7-day pass is Y24,000. Only going on the routes you described you would come out a bit ahead using the pass or about the same if you didn't get reserved seats. You can check yourself using hyperdia.com and put in your exact routes. The default is reserved seats. There is also a 4-day flexible pass for Y22,000 which lets you stop for a few days without feeling you have to travel.

And to answer your first question, yes, all those trips are covered by the pass.