Why do Yanks take their own families to court over retarded accusations?

Because if the dispute involves money, the court can order a family member to pay for damages or whatever owed to another family member.

That's pretty much the nature of any "normal" court case. Doesn't matter if it's between family members, former spouses, friends, etc.

I assume in the case of Judge Judy though, both sides get paid something for being on the show.

Americans like to sue for everything. Someone could drive drunk at night, run over someone and crash into a house and they'll sue the city for not making the street lights bright enough, and not having a strong fence to prevent him from crashing into the house.