I'm sorry to hear about Winston's raspy voice, wheezing, lethargy, hiding, gagging, and probable sore throat. Does he have a decreased appetite?
I know that he isn't sneezing but this does sound like an upper respiratory infection. Symptoms can range from a sore throat, loss of voice, lethargy, fever, nasal and/or eye discharge, sneezing and coughing. I suspect his change in voice and gagging are due to a very sore, swollen throat with increased amounts of mucous. These infections are easily picked up as they are spread by aerosol when a sick cat sneezes or through nasal or oral secretions when kitties share food and water bowls and litter boxes. Although your new kitten seems healthy it is possible that he/she was sick and was still shedding virus particles when he/she came to live with you. Just like a cold in us most of these are caused by viruses. Most respiratory infections clear up with supportive care and don't need any sort of medication. For the most part these viruses run their course in 7 to 10 days. There are some viruses however that are more virulent or aggressive than others.
If your cat has an immunosuppressive virus like Feline Leukemia or Feline Immunodeficiency Virus he may be sicker, longer or be more likely to get pneumonia and need veterinary care. Has he been tested for these viruses? Has the new kitten been tested for these as well? If not that would be something I recommend your veterinarian doing if his symptoms persist beyond a couple weeks.
Some of these viruses can cause severe ulceration of the mouth and gastrointestinal tract which will interfere with eating. If he isn't eating well to help reduce the amount of gastric acid being produced (which makes these ulcers more painful) you can give Pepcid ac (famotidine) at a dose of 1/4 of a 10mg tablet per 5 to 10 pounds of body weight every 12 hours. This may help him eat a little better.
It's also possible that he has a very sore throat and that makes swallowing painful.
Supportive care includes making sure he continues to eat. Spoiling him by warming some canned food or adding tuna or meat baby food to increase its smell and palatability is fine if his appetite is off. Encourage water drinking and add water to the food if necessary. If he won't eat and drink on his own at all after 24 to 48 hours of not eating you will need to use a syringe (medicine dropper) and force feed him and get fluids in. You can use canned food mixed with water or meat baby foods.
His throat may feel better if you put him in the bathroom with you when you take hot steamy showers.
If he starts to sneeze and his nose gets very congested you can use sterile saline nose drops to loosen the mucous and flush it out. He probably won't like it much but it will help him to breathe easier.
You can use sterile saline to flush the eyes if he gets a lot of mucous in them and artificial tears several times a day to keep them hydrated and comfortable.
Sometimes these viruses will become a chronic problem leading to on and off episodes of sneezing or eye discharge. Two common respiratory diseases are caused by Herpes virus or Chlamydia. You can give him an L-lysine supplement at a dose of 500mg orally twice daily to help shorten the infection's duration and it may curtail outbreaks by interfering with virus replication.
If he develops a discharge that is green or yellow that indicates a secondary bacterial infection in which case oral antibiotics will be helpful. He will need to see his veterinarian for those.
If he runs a fever more then 104F, stops eating entirely, or is very lethargic or has a bloody nasal discharge he should see his veterinarian.
Other possibilities for his symptoms are a nasopharyngeal polyp or a mass on his larynx so if he's not improving at all after a week he should be seen by your family veterinarian.
Let me know if you have any further questions.