I wouldn't worry. The only way it would be in trouble of being put down is if it attacked a human with little to no provocation.

If a family member killed an innocent human and was thus killed in vengeance - Yes 

If my pet killed someone else's pet and they in turn wanted mine put down - Yes

I don't believe in Ghandi's 'An eye for an eye only ends up making the whole world blind' nonsense, that just leaves one person blinded whilst the other faces no repercussions for their actions. I believe that justice in equal measure is necessary to restore equilibrium.

That you understand it is mostly your fault is good, but try a bit harder to remember that cats are pets just the same as dogs, and people become emotionally invested in them. 

While the majority of fault is placed on you, it is the responsibility of all pet owners to make sure their animals aren't in harms way, or accept that accidents happen. A hawk could've swooped down and killed the cat just as easily as your dog. When I take my parrots outside (which cost many times more than most cats and are just as if not more emotional and intelligent) I am 100% aware of all the things that can harm them, including cats. If your neighbor's cat killed my parrots do you honestly think he'd be alright if I chopped the cat's head off or suffocated it as a reaction? I mean some of my parrots cost a few thousand dollars, and feral cats are a MUCH larger problem than feral dogs, because people don't want to take care of their cats properly. 

Just as a cat can kill my parrots (especially my conures and budgies) and I would be at least partly responsible for their deaths, so should the cat's owner be at least (in a small part) responsible for the cat's death. Dogs kill smaller animals, even well-trained dogs will kill groundhogs and rabbits; and they won't discriminate against a cat. 

Fight for your dog, but make sure you learn from the lesson and take control of your dog. Either that or give the dog to somebody who can take proper care of it. Likewise your neighbor should not have been casual with his ownership over his animal. He should have known that there is a risk to having an outdoor cat; whether it is killed by a dog, a hawk, or a car. 

Did the cat owner witness the attack? How do we know that it wasn't some supper bad-ass cat and that your father's dog barely escaped with his life?

I have a cat. He is my little buddy. If a dog killed him, I'd be in a rage. If you knew the dog hated cats, you should have kept him inside and chained while outside. A dog attacked me when I was a kid too. My cat is like an extended family member. I understand if they want the dog put down. My suggestions are to beg for forgiveness, buy a chain and show them that you will use it, offer to do whatever you can, and mostly keep the dog inside for months (to help keep it out of sight and mind for awhile) unless being walked on a leash. Best thing to do is to give the dog away to a more responsible person, though.

Dogs are hunters and kill smaller animals. Some breeds more than others, but its normal. If your dog was a danger to people, it would have bit/attacked the cops, and they would have killed it on sight.

I think your dog will be fine, but you need to make sure it doesn't get out again.

You really should have been more careful with your dog bro. Try to work things out with the neighbor; they were probably just really upset in the moment. If they're an animal lover, they probably will agree your dog doesn't deserve to die for your/ your friend's mistakes.

By this token everyone with cats should keep their cats indoors or chain them, because cats hunt many more pets than dogs do and are much less trainable to not to. Do you agree that cats shouldn't be allowed outdoors unless they are on a chain? 

Yes cat's shouldn't be allowed outside, they're very hameful to native wildlife. Doesn't change the fact dogs are dangerous to humans while cats aren't. This isn't about the dog killing the cat but the chance it would attack a small child, which is a very high chance it would, it didn't kill the cat for food but for fun. It's a dangerous untrained dog and should be put down.

I hope nothing happens to your dog. It's not fair kill an animal due to his natural behavior/instinct. I think in this cases they should just sue the owner and make him/she pay $$$ for it. Nothing out of this world either, because everyone can make mistakes.