I do apologize for the glitch with my last response (it had been intended for someone else and seems to have posted to your page as well). In any case, I am very glad to hear that he hasn't any pain when you press on his stomach, but those sticky gums are a worry (since they are an early sign of dehydration creeping in). Now just to note since you mentioned he is also snotty and sneezing, that does actually raise some concerns of cat flu here. And I would just note that if that snot is yellow/green, we'd want to ring his vet about antibiotics.
Otherwise, I do want to note some other supportive options you can use to help here. To start, you can steam treating him to reduce his congestion. To do this, you can take him in the bathroom while you run a hot shower. The steam will help loosen and clear some of the snot congesting him. You can also use a baby nebulizer, but often cats don’t like things held up to their faces (but you can often make little ‘steam tents’ with him in his carrier, the nebulizer next to it, and a bed sheet over both).
Furthermore, since he has snot and discharge you can keep this wiped away with cotton balls soaked in warm water. As well, you can use saline nasal drops like Ocean Mist or Little Noses (but not anything medicated) to further reduce any nasal based congestion and discharge build up. To do so, just tilt his head back and drop two to three drops in one nostril. Cats hate this, but it helps. After the drops go down, you can let the head up and wipe away any discharge that gets loosened. Then repeat with the other nostril.
As well, if herpes virus is supect it can help to treat with L-lysine. This is a nutritional supplement that can help them recover quicker. This is available over the counter at pet stores, vets, online (even Amazon) and health food stores. This comes as gels, powders, or crushable tablets that can be added to food. An average cat dose is 500mg a day.
Finally, since you mentioned that he won't eat, do try warming his food or offering smelly wet food. When cats are congested and have sore throats, they often won't want to eat. So, this can be helpful in getting them to eat. As well, since wet foods are 35% water, it can help get fluids in even if his drinking is poor.
Overall, his signs do sound related to flu where the gagging could be a tracheitis with this. Therefore, we'd want to start the above supportive care and speak to his vet about potentially treating with antibiotics (as opposed to steroids) to see if we can get him back to himself.
Please take care,