Usually people with cat allergies have a runny nose when in a room with cats. what happens with people with allergy to dogs? do they ever have skin problems? someone just returned a rescued dog saying that their son had a rash on the upper body and neck because of the dog, nevertheless he never had shown problems with dogs just being in the room. he is allergic to cats, but with the usual runny nose when in the same room. does their story sound plausible or is it an excuse to get rid of the dog?

That can be a reaction. Some people will get hives being in close proximity to a dog or cat while others have to actually touch them. Animal dander is an extremely irritating allergen because of all the barbs on it.
I have terrible allergies and take prescription serum for it, luckily I'm not allergic to animals, just everything else in nature…lol. However, my friend is allergic to both, and he will break out in hives if he's too close to a cat, or touches a dog with anything other than his hand.