"After the boy fell, his knee started to bleed."
Translation:Post kiam la knabo falis, lia genuo komencis sangi.
Because "his knee" is the subject of the second clause ("started to bleed") and "sia" is never used inside the subject.
"After the surgery" is "post la operacio", but "after the boy fell" is "post kiam la knabo falis". Why does "kiam" appear in the latter but not the former?
Because "the surgery" is a noun phrase which can take a preposition but "the boy fell" is a clause (with a verb) and so needs a conjunction.
If it helps, you can try to replace "before/after" with "because", because there English makes a difference between conjunction and preposition: "because the boy fell" but "because OF the surgery".
So if you would have to say "because of", you would use just "antaŭ/post" in Esperanto for "before/after", while if you would have to say "because" without "of", it will be "antaŭ ol/post kiam" with an additional word in Esperanto.
No, because -igx- has nothing to do with starting. A case could probably be made for eksangis, though.
Because that would mean "Afterwards, the boy fell, ...".
poste is an adverb, not a conjunction.