When a child is adopted, they become connected in the family tree as the children of whoever adopted them, there is no mention of the child's history of being adopted. As far as getting the parent's genetics themselves, I can't see any resemblance, unless they already shared some similarities such as hair, eye, or skin color. I haven't had too many sims adopt children, but I have a couple of different times, and this is what I noticed. As the children aged up, they didn't share any of the biological traits as their siblings. One son once he became a teen he was extremely muscular, even though he never worked out, while his parent's biological children once they aged up were thinner and fewer muscle mass. Another mother who adopted a baby girl, when that little girl aged up she had weight issues, while her siblings were skinny toothpicks.
As far as grandchildren I have less information on that, because of the above sims only one of those had one grandson.