I know a professor of mathematics that makes bows as a hobby (the kind that shoots arrows). He made a LOT of them so far. Apparently, he still finds it fun. Eleazar, have you ever actually had a hobby like that?
"But Judea Pearl tells me just how to compute a counterfactual, given only my beliefs about the actual world."
This is actually a subtle issue. The procedure given in the book assumes (a) full knowledge of the precise causal mechanisms (which you never know in practice) and (b) the distribution over all unobserved variables (which you don't know by definition). Surprisingly, it is possible to compute certain counterfactuals using ONLY the distribution over observable variables (which is typically what you get). You can check my thesis for details if you wish.
About choices: what is the criterion by which new ontological primitives are to be added?
If the universe is timeless, but causal, it is an interesting empirical observation that causal direction never seems to contradict 'temporal direction.'
I don't want to speak for Pearl, but my understanding of his position is that causality is more fundamental than probability in the human mind (not necessarily more fundamental in reality).