You mention only top sites in Hokkaido. Haven't you been some of there on the last trip?
Those sites are well covered by public transportation. Driving trips should be constrained on only inconvenient areas. I should say driving could not save your time. You need to slow down the trip in driving tours. A faster pace with public transportation or slower pace with driving. You could take a rest on each ride of the public transport, while you could not do when you are driving. If you dare to choose a faster pace with driving, you should hire a driver for that. The driver could take a rest while you are walking around at each site. Otherwise, it could be a road to the Heaven.
Here are driving time estimates by Google Maps.
2:16 Hakodate - Toyako
0:48 Toyako - Noboribetsu
1:10 Noboribetsu - Sapporo
0:25 Sapporo - Shiroi Koibito Park
2:10 Sapporo - Furano
0:44 Furano - Biei
2:24 Biei - Sapporo
0:46 Sapporo - Otaru
In travel, traveling itself is your work. It would be better to limit your travel to eight hours a day. Eight hours of a sleep, another eight hours for others; such as breakfast, dinner, bath and so forth.
For a more satisfying trip, the transportation time should be less than half; up to four hours a day. Then, The answer to your question #2 is clear. You should not do a day trip to Furano and Biei from Sapporo. It would take more than five hours for only the loop of Sapporo - Furano - Biei - Sapporo.
My rule of thumb for a driving trip is to give a day per site. A set of driving, sightseeing, and sleeping for one place. Then, I would consider something like this.
A: 5 night 6 days
Pick up the least five populated area. In this case, Furano, Biei, Lake Toya, and Noboribetsu for the least four. Only one city stay can be allowed. Asahikawa airport is very close to Biei. Check in a car at Asahikawa airport and check it out once you arrive at the last city. You could check out the car at the last airport, but I think, in a city, the convenience by a car would easily be destroyed by difficulties in finding the route, parking lots and so on. The inverse direction is OK, or desirable to avoid a tiredness after a flight. But a start from the city might be suffered from the difficulties above. For Hakodate, you could use a Shinkansen train from Tokyo. Note it is just a one train ride from Otaru to Chitose airport.
A1: Asahikawa airport - Biei - Furano - Noboribetsu - Lake Toya - Hakodate - Hakodate airport
A2: Asahikawa airport - Biei - Furano - Noboribetsu - Lake Toya - Otaru - Chitose airport
A3: Asahikawa airport - Biei - Furano - Noboribetsu - Lake Toya - Sapporo - Chitose airport
B: 6 nights 7 days
Two cities are allowed. B1 is an easy extension from A2 and A3. From A1, I create B2 and B3. Shiroi Koibito Park is on the way to Sapporo from Otaru, or one subway ride from Sapporo Odori metro station, or one bus from JR Sapporo.
http://www.shiroikoibitopark.jp/english/access/B1: Asahikawa airport - Biei - Furano - Noboribetsu - Lake Toya - Otaru - Sapporo - Chitose airport
B2: Chitose airport - Sapporo - Biei - Furano - Noboribetsu - Lake Toya - Hakodate - Hakodate airport
B3: Chitose airport - Otaru - Biei - Furano - Noboribetsu - Lake Toya - Hakodate - Hakodate airport
C: 7 nights 8 days
All the three cities you mentioned are allowed. C1 from B2 and B3. This would be the most similar one to yours but needs 7N8D. In this scheme, you could not gain a time by car in the portions of Chitose airport - Otaru - Sapporo and Lake Toya - Hakodate - Hakodate airport. More precisely, Chitose airport - Otaru - Sapporo - Asahikawa and JR Toya - Hakodate. So, checking in a car at JR Asahikawa and checking out it at JR Toya is the most favorable plan.
C1: Chitose airport - Otaru - Sapporo - Biei - Furano - Noboribetsu - Lake Toya - Hakodate - Hakodate airport
If you are a person of day-trips, stay in Sapporo, do Otaru from there. Stay in Furano, do Biei from there. Stay in Noboribetsu, do Lake Toya from there. In this case, check-in at Sapporo just before to Furano, and check out at Noboribetsu just before to Hakodate. Such as
D1 Chitose - Sapporo, 0.5-day Sapporo, Shiroi Koibito
D2 1 day Otaru from Sapporo
D3 Sapporo - Furano via R452 by car, 0.5-day Furano
D4 1 day Biei from Furano by car
D5 Furano - Noboribetsu via Shimukappu by car, 0.5-day Noboribetsu
D6 1 day Lake Toya from Noboribetsu by car
D7 Noboribetsu - Hakodate, 0.5-day Hakodate
D8 Hakodate - Tokyo?
Lake Toya should hold fireworks at night in summer. You could switch the night at Noboribetsu for that.
