What is the best way to get around without a car?
Public transportation, including train, bus and taxi.
Is hiring a chauffeured car or taxi just completely out of the question?
No. I certainly don't recommend it for the entire stay, but for some parts of the trip, where public transportation is inconvenient, it might be a good option.
Do buses actually put you near where you need to go?
Sometimes yes. Sometimes no. It varies case by case.
Start at Sapporo, then travel to Noboribetsu for 1 night and Otaru, after which we'd like to get in some time at: Daisetsuzan Park (stay at Ashidake Onsen for 2/3 nights)
then back to Sapporo and leave.
Certain hotels in Noboribetsu provide a free shuttle bus service to staying guests between Sapporo and Noboribetsu. That would be a very convenient way of travel. For example:
http://www.takimotokan.co.jp/english/access/index.html
Inside Sapporo, however, it is probably better to use taxis instead of buses and subways, because walking and some stairs is often involved when using the subways and buses. Especially, in case your mother is not be so strong at walking, anymore.
Otaru is relatively easily accessed by train, but for medium and longer distances within Otaru, a taxi might again be the best way of getting around.
As for Asahidake Onsen, note that besides hiking there is not much to do around the mountain resort. Nor is it a convenient base to explore the surrounding sights. I think that one night is enough. Two nights at the most. Access by train and bus via Asahikawa is quite convenient, but look up the timetable in advance, as bus service to Asahidake Onsen is infrequent.