"Pozwolić" is perfective, "pozwalać" is imperfective. Perfective cannot be used in the present, only in the past or future tenses. It shows that the "allowing" is complete. "Pozwolili" = they allowed, it's allowed, the end. That's probably not a very good verb to show the perfective/imperfective difference on. But then again, being level 11, you probably already know quite a lot about it :)

Therefore "pozwolą" would mean "they will allow", while "pozwalają" is "they are allowing right now" or "they do allow generally".

Because "pozwalać" (to let) takes Dative, and Dative of "dzieci" is "dzieciom".

I'll make the default English version "their children", actually.