On the topic of self-driving vehicles: I absolutely would trust a self driving car. Nearly all accidents are caused by user error, either from distracted or impaired driving, and computers just don't make those kind of mistakes. Google has been testing its cars in california for a few years, and the ONLY accidents they've been in so far have been either someone else crashing into the self driving car (usually rear-ending) or when the "driver" of the self driving car took over and the car wasn't actually self-driving.
Your "intuition" can absolutely be replaced by actual calculation. A car can have cameras and sensors all the way around it so it can be far more aware of the road than a person ever could be. Even if a computer does end up making a mistake driving (again, which hasn't happened yet) it is far less likely than a human making the same mistake. I am thrilled for a future when people who are impaired, whether from alcohol, cannabis, sleep deprivation or anything else, can safely get home without putting others in danger.
Would I trust a plane without a pilot? Absolutely. Planes are in general much safer than cars already, and the few plane crashes I can remember in the past few years were caused by user error.
Granted, all this is years and years away since the tech is moving far faster than the legislation. The CA-imposed law that self driving cars require a sober driver ready to take over is a big problem in my opinion but I'm sure it'll be changed once self driving cars are more proven. As is, it prevents these cars from helping the people who really need them: the disabled, especially the blind. Then after all that the price has to come down to something reasonable, but I don't think that'll take as long as you'd expect since most of this is software and the hardware cost of the sensors needed isn't going to be a massive cost or anything.
As for the "mowing down children or hitting a wall" the wall is absolutely the correct choice in that scenario. The kids have no protection from the car, which will very likely kill them. The wall MIGHT kill you, but you have a better chance since you're protected by a crumple zone, seatbelts, airbags, etc. So if you hit the wall there's a much better chance that everyone survives. If you'd choose to run the kids over, sorry but that's pretty fucked up. The car isn't "deciding" to kill you, it's trying to minimize the chance of anyone losing their life.