How will it react to a bicyclist running a red stoplight and into the car's path, when the car has a green light?
It will stop. In fact, the "failure mode" of self-driving cars will be how pedestrians can bully them by stepping in front of them so they stop.
When the car comes across kids playing football/street hockey/baseball in the street, will it stop for them?
There is already video of this. The radar/lidar isn't good enough to see a baseball, but it's definitely good enough to see a kid.
Will these cars have systems to react when they take damage?
It will probably pull over and stop if something is wrong.
https://www.ai-class.com/course/video/videolecture/48
https://www.ai-class.com/course/video/videolecture/209
https://www.ai-class.com/course/video/videolecture/210 (Lombard Street at about 4:40)
One thing that strikes me is how much more these robots can potentially see than a human driver. Imagine how much better you could drive if you could see in all directions at once with multiple range-finding sensors and cameras, track dozens of moving objects simultaneously, never get sleepy or look away from the road at the wrong time, react to events in milliseconds, and learn from hundreds of thousands of miles of training data as your driving "experience".
I think the answer is: "not very well."
How would it react to a roadway that's covered in several feet of water due to flooding (and poor road design)? Or sinkholes? One-way streets? Freeways which change directions at different times of day? Dirt roads? (Can we make it autonomously offroad?) How will it react during an earthquake or tornado? How will it react to a bicyclist running a red stoplight and into the car's path, when the car has a green light? When the car comes across kids playing football/street hockey/baseball in the street, will it stop for them? How far out of the way will the kids have to move before the car thinks it's safe to go past?
Will these cars have systems to react when they take damage?