I haven't read all the posts but isn't leaving a dog outside in the heat like that animal cruelty? Also I would think that if they have already had animals taken from them that there is probably some sort of stipulation about them having another dog. I think I'd call it in again to file a complaint and ask them what the rules are on that.
if they aren't home, I'd offer the dog some shelter from the heat. Yes, I'd take the dog to my house and then discuss it with them when I returned the dog. If I returned the dog.
Maybe they'll think it ran away.
p.s. don't forget to mention that you have friends in the 'mafia'. LOL