First, let your vet know that you are not comfortable with the "wait and see" approach. Be very clear about your feelings. If he/she does not respect that and agree to try to get to the cause of problem immediately, find a different vet. Now, let me try to answer your specific questions.
Pemphigus is an autoimmune disease that is indeed very serious. These dogs will develop sores at the mucocutaneous junction, that is, where the skin meets mucous membrane-type tissue. This includes in top of the nose and also on the lips and around the eyes. It can also be around the rectum or around the sheath of the penis. It is diagnosed by taking a biopsy of the affected skin. This can be a pretty serious disease but it is treatable. Some dogs do not tolerate the treatment well or can develop complications, so although it is not generally a fatal disease, it can be expensive and complicated to treat.
There are several over possibilities besides pemphigus as to what could be going on with your dog. Because you mentioned a cough, doing chest x-rays may be helpful. One of the things that could cause both the cough and the sores on the nose include systemic fungal infections. Which one is most likely will depend on what part of the country you live in but they include Blastomycosis, coccidiodomycosis, cryptococcus, and histoplasmosis. It is likely that the combination of chest x-ray and biopsy of the nasal lesions could give you the diagnosis.
There are other fungal type infections that affect just the skin that could cause this. Also, on dogs with light colored coats, they can develop a reaction on their nose to sunlight called nasal solar dermatitis. Also, some types of cancer (also usually in light colored coats) can occur here.
There are other autoimmune diseases that can cause the lesions on the nose, including systemic or discoid lupus erythematosus.
Dogs can get a condition where there are mutliple warts present in and around the mouth. While this would not typically occur on top of the nose, if there are similar lesions inside the mouth this could be causing what you are seeing.
It is likely that a biopsy of the area will be necessary to get an absolute diagnosis for you. Be sure to mention the coughing to your vet as the biopsy will require anesthesia and we need to know what is going on in the lungs before the dog is put under anesthesia. It is possible that they will want to do some blood work as well. Once you have a diagnosis, you can discuss the treatment options available with your vet and decide what is best for you and your pet.
I hope this answers your question about what could be causing these lesions on your dog's nose. If so, please click Accept. If you have additional questions, please let me know.