Can a cat give a dog a disease? Sorry shoud be in GQ.
Well, there’s always rabies.
As far as I know, dogs and cats can freely exchange rabies, tetanus (rare), and a number of parasites.
Diseases that can be transmitted from one species to another are called “zoonotic” diseases. For example, there’s a zoonotic bacterial infection called “leptospirosis” which can be transmitted from cats to dogs and vice versa. It can also be transmitted to and by humans. Link to an article about leptospirosis in cats: Bacterial Infection (Leptospirosis) in Cats | PetMD.
So moved, from IMHO to General Questions.
Worms.
I am going to guess that cat > dog is more common than dog > cat. Due to the presence of [del]cat poop[/del] chewy yum patties.
Are the worms which afflict dogs the same species as those for cats?
Uncinaria stenocephala at least.
Some of them. It really varies widely. You can have generalists like the tapeworm Dipylidium caninum or semi-specialists like the roundworms Toxocara cati and Toxocara canis.
A lot of them prefer one host or another, but can make a jump to another. Heartworm is a classic example. It is far more common in dogs, but can be found in cats.
Meanwhile humans get pinworms, but dogs and cats don’t.
Fleas, ticks, and scabies (Sarcoptes scabiei) can also be freely shared between cats and dogs. Same with ringworm, which is actually a fungal infection. In fact, the cat may be asymptomatic and only the dog show clinical signs.
This is the reason why, in the “my dog/cat has parasites, what do I do?”-threads, part of the advice is to treat both the animal known to be affected and all the other critters that so far don’t look like they have fleas/ticks/parasites.
Also, if a cat attacks a dog, the dog can get an infection with Pasteurella multocida. But I’m guessing that’s not the type of disease transmission you’re referring to. 
Actually, no. Zoonotic diseases are those that can be spread from animals to humans.
From: Representative medical dictionary: