For mechanical failures, such as flat tires, the car could just pull to the side of the road and fire off a text message to you and AAA (maybe we require you to have something like a AAA membership if you want your car to drive itself). They come and fix the tire, tow you to an auto shop, or whatever needs to be done. You're inconvenienced, but certainly no more than you would be otherwise.
As for cultural changes, here's what I imagine will happen: At first, people will be uncomfortable with self-driving cars. As another commenter noted, people don't feel safe with things they don't control. But most users of the cars will need very little time to get used to it. They'll hop in their car, tell it where to go, and proceed to pay no attention at all to what the car is doing. Once enough people have had this experience, it won't be difficult to convince the public to allow the cars to strike out on their own.
As for cultural changes, here's what I imagine will happen: At first, people will be uncomfortable with self-driving cars. As another commenter noted, people don't feel safe with things they don't control. But most users of the cars will need very little time to get used to it. They'll hop in their car, tell it where to go, and proceed to pay no attention at all to what the car is doing. Once enough people have had this experience, it won't be difficult to convince the public to allow the cars to strike out on their own.