It depends who you talk to ... My vet has noted dog to cat URI s and vice versa but in general most will say a dog cannot give most ailments ( rabies I believe is a firm yes) to a cat
Do you mean "diseases" in the literal sense? There are some I am sure but I can't really think of any at the moment, but there are a bunch of things that would fall under other conditions, most parasites external & internal. I wonder about some bacteria but I don't know. My vets insisted that Clostridium was species specific, but I proved them wrong, of course my gang loves to prove the Docs wrong