I assume when you said asahiyama you mean asahikawa? There is asahiyama zoo but it is located at asahikawa
http://www.japan-guide.com/e/e6890.htmlMy general recommendation is as follow:
Fly in new chitose, rent a car there, drive to asahikawa/furano/biei, return car at sapporo, spend your time in sapporo/otaru, take train to noboribetsu/toya/hakodate, fly out of hakodate.
For your specific questions:
Is it worth renting a car for the whole trip? Its quite expensive for car rental. Or should I rent a car when doing Asahimaya, Furano and Biei? If so, should I rent the car from the Airport or take train to Asahimaya and get the car there is cheaper?I would recommend renting car for the part of the trip I mentioned above.
Is it easy to visit - Daisetsuzan National Park with 1 year old baby? Which area should we visit if we have limited time? Could one day cover Furano and Biei?This really depends on your pace. You could just spend like 30mins on each field if you have flower or more if you like to enjoy the flower field. I would skip Daisetsuzan or put it as optional in case you have more time. It takes about 1.5hr by car depending which part of daisetsuzan you are going and more by bus.
Would there be anything in Asahimaya Zoo during summer or shall we pass it?Not sure I understand your question. It is a zoo so there is animal all year round. If you like to see animal then it is worth it.
Should we stay in Furano instead of Asahimaya?Probably better at furano but it is not terribly far from asahikawa (1.5hr by car)
Can we do day trip to Shiretoko? Is it easy access with one year old baby to go in a pram?It takes 4.5hr by car from asahikawa to get to shiretoko so day trip is almost impossible.
How many day would we need in Hokodate to visit attraction places?It depends on your pace but 1-2 days is the norm.
Is it possible for a day trip to Lake Toya and Noboribetsu with public transport and a baby from and back to Hokodate?Again, depends on your pace but it s very tight as the transportation isn't very frequent.
- Is 3& half days more than enough for Sapporo or should we visit somewhere else as well?Again depends on your pace, but I will recommend spend take out a night and spend it on noboribetsu/toya
- Any cheap car rental company? Do they give English GPS? Map?http://shiretoko-t.com/en/
http://www.japan-guide.com/e/e2024.htmlThere are some links at the bottom.
Should we get JRpass or Hokkaido free pass?Neither if you follow my recommendation. Sapporo-Otaru pass is recommended
http://www2.jrhokkaido.co.jp/global/english/railpass/welcome.html