So, these are quite a few unrelated issues here.
Firstly, a video would be helpful to determine what is going on with his respiratory system. We may consider some basic tests such as a chest xray if concerned. Some Frenchies benefit from surgery to help widen their airways.
Itching can have many causes including allergies, bacterial or yeast infections, mites etc. Some basic skin tests such as a skin swab and scrape are usually the first step and he may require some anti itch medicine to settle things down.
Your partner may have a dog dander or saliva allergy and should discuss this with their GP. Parasites such as mites can be infectious to humans but there would be skin lesions.
Flea collars (unless you mean the Seresto collar) are rarely effective. Spot on such as Advocate are best. Ensure all pets are treated, as well as the home.
With regards ***** ***** this is serious and needs to be addressed asap. It is possible it is driven by the discomfort of his itching as any medical issues can mean a dog is less tolerant of people and animals. I would firstly address his medical issues with his vet and, if the behaviour continues, seek assistance from a canine behaviourist who can assess him and provide a tailored plan.
Ensure he is getting plenty of mental and physical stimulation (in the form of walks, training, food puzzles, interactive toys etc) and consider some calming supplements and a calming collar such as an Adaptil collar.