I may not have given enough detail. The prohibition against killing is specifically innocent people. There is a death penalty for many crimes, including murder (although not as far as EY seems to think. He once said that the Bible gives the death penalty for crossdressing. Evidence suggests otherwise. But that's another topic.) So:

So, if you see your children being slaughtered in front of you, and all you need to do to save them and to kill the attacker is to press a button, you are not allowed to do it?

Assuming this attacker is the one killing or threatening to kill your kids, you are allowed to kill him (although you are supposed to try to injure them if killing isn't necessary to stop them). You wouldn't be allowed to kill someone else who is innocent, even to save many people.

Also, does this mean that there cannot be Orthodox Jewish soldiers? If so, is this a recent development, given that ancient Hebrews fought and killed without a second thought? Or is there another reason why it was OK to kill your enemy in King David's time, but not now?

I don't know if you're familiar with the current debate in Israel over the draft? It's not really related, though. Again, the “ancient Hebrews” fights, were usually either to reclaim parts of Israel which belonged to them from the gentile nations that were inhabiting them, or to defend themselves against attackers. In both scenarios, the “victims” weren't innocent. For some more info, see here, here, and here.

(By the way, I just saw this while looking up that last link, which (mostly) confirms what I said about the Trolley problem.

is this a utilitarian calculation or an absolute injunction, like in the previous case, where you are not allowed to kill, no matter what? Or is there some threshold of (dis)utility above which lying is OK? If so, what price demanded by the selfish driver would surely cause a good Orthodox Jewish hitchhiker to attempt to lie?

I realized after I posted that answer yesterday that I could conceive of a case that would work for me, in the spirit of the Parfit's hitchhiker example. Namely, if I knew that when I got to town there would be someone who's life I could save, but only with $100. (Also assuming that I've got only $100 cash total). That person's life would take precedence over telling the truth, and I wouldn't get the ride. There isn't anything I could do in terms of prior obligation that would override the life concern of that person later.