You mentioned that the children of a Cohen and a divorced woman are Challalim. Please explain that status, as my children are growing up alongside children of such a marriage, and I should know the implications of their status in case my child wants to marry one of them.

A Cohen is not allowed to marry the daughter of a Cohen and a divorced woman (Shulchan Aruch Even HaEzer 6 1). When this daughter marries a Jew who is not a Cohen, their daughter- the grand-daughter of the Cohen and the divorcee- can marry a Cohen (Shulchan Aruch Even HaEzer 7 16).