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It looks like every Kid rolls once, adding dice for attribute, corresponding skill, and items they may have in their possession.

They get to reroll all the dice that failed by spending a Luck Point, once per test.

And after that, if they have at least half the successes they need to pass the Extended Trouble, they can suffer Conditions for free successes.

Do I have that right?

Also, using Lead to build a dice pool for the other kids instead of contributing successes to the Extended Trouble seems counter-productive, like the leader's time would be better spent doing things instead of leading.

I think so, with the exception that the players can use both Luck and Push their rolls. Is it worth to spend a condition now for the chance of more successes?

Now I'm looking at this and thinking about strategy.

For any given roll they can both spend Luck and Push (thereby taking a Condition) to reroll the failures in their dice pool. Each die rolled this way has a 1 in 6 chance of becoming a success.

But then, as part of the Extended Test, they can trade Conditions for successes, one for one. Depending how many dice they have to roll, and how many Conditions they've already taken on the way to the climax, this might be a better deal than using the Push.

Absolutely, its about risk and reward. Will they gamble?

Also, using Lead to build a dice pool for the other kids instead of contributing successes to the Extended Trouble seems counter-productive, like the leader's time would be better spent doing things instead of leading.

I’m struggling with that passage too: if one kid would have done a Lead Test earlier in the Session/Mystery it would be clear that this kid could also contribute to the Extended Trouble. Then it is stated that the Lead Test can only be done if there’s enough time.
My reading is that one kid may do a Lead Test to build a pool (but the successes from this test may not be used to contribute to the main task) and after that all kids (including the leader) make their contributing tests.

Most of the time, the best use of the PCs is to have them all participate in the various skill rolls for the extended trouble, but who knows, maybe in a situation the PC with 'Lead' doesn't have any useful skill for a particular trouble, but the 'Lead' PC is very good in just 'Lead', then it could be a good way to at least contribute something.

But the 'Lead' skill is best used for normal troubles, that's true.

Thanks for the answer and I'm afraid you are right: in the (german) GM Screen it is clear that a leading kid has no action to contribute.