Question: Who exactly are "the new founding fathers"? It never actually says. I read an internet post that said they were not singular people, but a collection of multi-national companies that took over the government after a near nation wide collapse of the US.
Question: Why didn't those purge people just go find someone else? Why waste all their time on one house and why did the girls go around in flimsy dresses and barefoot on a cold night?
Answer: These purgers are pretty crazy. They were hunting this person, a homeless man, and they wanted to kill him specifically. They wanted their prize, the trophy of their hunt. And they won't let anyone stop them as it's their right. So being blocked from their target only increases their desire.
Answer: Shortly after letting the black guy into their house, the gang leader tells James that he (the black man) killed one of their members, and they wanted revenge.
Answer: The only information the original script offers is the company that took over the government, their name is "Arcon". Other than that, there is no other specific name nor is there an answer to what kind of company Arcon really is.