If he’s passed a stone before it’s likely he may have another. Male cats have a really hard time passing stones or crystals since their opening it very small. Was the cat showing signs of a uti, or was this routine labs? Do you know if they looked at the urine to check for crystals? Sometimes it could be an infection and a round of antibiotics clear it up, but id be weary of that diagnosis if they didn’t look at it under a microscope.

Male dogs humping things could be because he isn’t fixed, and if the new puppy is female, yes he will try and hump, maybebif it’s a male too. I would definitely get them both neutered or spayed and that could solve your problems. Some male dog will continue the behavior afterwards, it just depends on the dog.

Has he had vaccines? Chew on toys? I agree one more vomit and take him to vet.