[susan53]

Here are some ideas regarding (B)

The children dashed/scuttled/scurried/sprinted/scampered off as soon as the headmaster appeared

dashed is fairly straightforward. It just means to go very quickly, in a hurry. Other examples : When we heard all the noise, we dashed out to see what was happening. / I'll have to dash off to catch the train as soon as the meeting ends.It's often used to apologise for leaving somewhere early, or not speaking to someone when you meet them : I'd love to stop and chat, but I've got a lesson. I must dash. (Heavy stress on must)
Used transitively, it can also mean to write something very quickly: I dashed off a quick answer to his e-mail before I left, but I'll write more fully when I get home.
And finally it can mean to destroy by throwing quickly and violently: He picked up the vase and dashed it to the ground. This can also be used metaphorically : When John refused to participate, he dashed all our hopes of winning.

Sprinted gives the idea of a short but very fast run (often a race), which starts very suddenly. In athletics, the 100m and 200m are sprint races. So an example might be : He saw the bus coming and sprinted to the bus stop. I don't think I'd use this in your example if the kids involved were very small, but with older ones, possibly.

Scuttled and Scurried are very similar to each other. They both mean to move with very small steps and movements, like a small animal :The hamster scurried around the cage looking for wisps of hay to build its nest / The porcupine scuttled across the road.
Scurried can be used metaphorically : His mind scurried frantically, seeking a solution to the problem.

Scamper is similar - it's associated with small animals (or children) and implies small movements. But it doesn't have the sense of speed or urgency of the other two: While the adults sat and talked, the children scampered around them playing games. In your context, it gives the idea that the children thought that escaping from the headmaster was a bit of a joke, rather than that they were in any real danger of punishment.

So here, they could all be used, but they create different mental images of what the "running away" was like.