Hmmm ... might suggest you talk to someone who has been involved in divorces and child-custody cases to get your mind straight in that matter.
I qualify. Are you asserting that the HOA case will be held in Family Court?
No, don't bother. It's obvious that you have strongly held, but erroneous beliefs. There's nothing gained by me refuting you point by point, because you're obviously here to make a point, not learn.
Children cannot be a party to a contract, because minors can't be held accountable for a contract. Any contract signed by a minor is null and void on it's face. They cannot, therefore, have standing as a party to a contract. The child is asserting no right to live there, the GP are resisting being compelled to uphold a contract they made.
Family law, on the other hand, carves out a legal sphere where "the interests of the child" are considered. The child is not a party to the marriage; the court considers all issues with the best interests of children in mind.
No, this case would not be heard in Family Court.