How does Stephen Hawking conduct his research?
Have you read his article in the Wikipedia? I believe most of your questions are answered there. 

As for students and researchers with disabilities in general: Every disability is different and different things can be done to help, from extra time on homeworks and exams, to special reading / blackboard viewing devices, hearing aids, keyboards that require very little finger strength, to having another person take notes for the disabled student / researcher. I'm sure I forgot lots of other tools. 

The most important tool for research is the mind and with a keen mind most obstacles can be overcome.