Question: When Dr. Strange gets into the accident near the beginning of the film, why didn't his car's airbag deploy? Wouldn't such an expensive car have all kinds of standard safety features?
Question: Why didn't Strange heal his hands and continue being a surgeon?
Answer: Medically, he could not heal his hands. He spent his entire fortune trying to return to being a surgeon through experimental procedures. Once he completed his training to become a sorcerer, he realised there was more to the world than he realised. The Ancient One correctly stated that he was a surgeon for his own ego; he wanted to prove he was the best and helped people for that reason only. By joining the sorcerers he was given a greater purpose and chose this over returning to his former life.
His hands were healing, just not quickly enough for his needs. This was showing his egotistical attitudes.
Answer: For now, he is still in the midst of studying the mystical arts that he has not yet mastered and may not be able or interested in returning to his former profession at this time. He has also assumed the role as caretaker of the New York Sanctum, and is dedicated to helping protect Earth (along with the London and Hong Kong Sanctums) from threats by the other dimensions.
Question: When Strange is surgically removing the bullet from the patient's brain, why did he ask the one doctor to cover his wristwatch?
Answer: I took it to mean Dr. Strange could hear the watch ticking, and he wanted complete silence.
Chosen answer: The airbag did deploy. However, the deployment occurred when the car first crashed and we see the accident from the outside, so we can't see it. Once we get a shot of the interior, you can see the airbag is sticking out from the steering wheel, but (unlike what some movies would show you) airbags deflate very rapidly after a crash, so you may have missed it.
But if the airbags did deploy, then why did he have such major injuries? Wouldn't the airbags have made what happened to his hands less severe?
I don't think so. I watched it again recently and I think I saw the car's speedometer crush his hands while they were still on the steering wheel, while the airbag deploys from the centre of the steering wheel, nowhere near his hands.
Often in real life, when air bags deploy from the steering wheel, they force the hands off the steering wheel resulting in the hands hitting the windshield. Damage to hands, wrists, and lower arms are common.