You'll hear from lots of people on this forum that dog parks are bad, but I go nearly everyday with my dog, now 16 months old, and have never had a bad experience. She's gotten in a few disagreements there at times, but has always recovered well and has not held any grudges. My dog plays well with large and small dogs, and does not bully anyone, except for dogs that act overly timid. And from what I've seen repeatedly, the dogs that get that unwanted attention quickly stop acting that way and the other dogs then quit picking on them LOL! Though it's sort of counter intuitive, it does seem to be a universal sort of dog thing. Anyway, glad to hear you've found a way to get your puppy some outlet for his energy!
And on the cat question, I don't have cats now, but when I did I let the dogs and cats figure out for themselves what worked and what didn't...within reason LOL! We had one kitten and a puppy one time that when watching them I thought the puppy was killing the cat. But when I intervened, because the puppy literally had the cat's whole head in it's mouth and was dragging it around, I was shocked to find that the cat was purring! They were inseparable their entire lives...not saying you should ignore bullying, but animals have their own ways that we often just don't get. Be watchful and stop any prey-like behavior you see, but beyond that, IME dogs and cats speak pretty well with each other, and they all have a way of working things out pretty well!
It's not the size of the dog in the fight, it's the size of the fight in the dog. Mark Twain