I could answer some of your questions:
1- We were thinking of staying at the King's Island Resort and notice they have a bus to transport back and forth to the park. I know that it'd be cheaper to do that, but does anyone know where they drop off at and/or if they transport scooters?
They drop you off at the passenger pick up/drop off area and I think they transport scooters.
2- Also - does King's Island allow you to bring your own scooter to the park?
Again I think yes
3- Do they have handicapped-accessible parking?
Yes, they have a whole section
Just call the park here: 513-754-5700 They should answer the rest of your questions
The number for KI Resort is (513) 398-0115
I forgot to mention that there are live shows, and bunch of other stuff you can do if you are a non-rider.