There is no reason to take a taxi from any resort to any theme park. Disney buses start running about an hour before theme park opening so you will get there in plenty of time. Disney also runs earlier buses to take people to the various character breakfasts, tours, etc. going on before park openings. We have never had any problem getting to any theme park before opening by using Disney transportation. Also, FYI, if you took a taxi to the MK you would be dropped off at the TTC and have to either take the boat or monorail over to the MK. Only Disney buses are allowed near the Magic Kingdom. They drop you off very close to the MK entrance.
As for your timing question, I would guess that the longest bus ride is maybe 20 minutes, with most less than that. The AKL would not share a bus with any other resort so you will have no problems taking the buses to all the parks.