Welcome to Just Answer. I'm sorry to hear your little one is not feeling as well as she could. Is she still eating and drinking normally? Is she having any nasal discharge or eye discharge? Does her cough sound like a goose honk?
Ok. I'm glad to hear that she's still eating and drinking. I'm very suspicious that your little one has kennel cough. That is a viral/bacterial upper respiratory infection of the trachea that causes irritation and coughing events like your describing. Here is a video of a dog with it.
If this is what is going on, then her body is fighting of that infection. If there is no nasal discharge then most likely it is viral at this point. If there is discharge then there is a secondary bacterial component. These are cases where I will recommend antibiotics to be given. If the coughing is persistent and she can't get comfortable, then cough suppressents can be used. These should both be obtained from your vet.
Here is a great link on kennel cough.
I don't think this is related to the skin issue. The other thing it could be is a collapsing trachea - but this is usually a small breed dog issue. Here is a link on it.
I hope this helps and your little one starts to feel better here soon. PLease let her rest and don't stimulate her. No smoke or dust as that can make it worse. Also no leash walks - use a harness.