I am sorry to hear that Snowball is drooling, walking around with his mouth open and not eating or drinking much.
Drooling can be related to eating something bitter (a bug or plant) or bad tasting, having something caught between his teeth, nausea, or mouth pain from gum disease or bad teeth.
Tooth and gum disease, a mass in his mouth, or organ failure with secondary nausea are other possibilities.
If he goes outdoors unsupervised trauma to his jaw/face is a possible cause, but I would expect you to report scrapes/marks on his face.
I would look closely into his mouth if you can to see if you can see redness, ulcers or anything caught.
If he has been sneezing and you see any redness or ulcers some upper respiratory infections, like Calici Virus, can also cause ulcers and these are painful, so they do cause drooling, and that is definitely a possibility in a cat that is excessively drooling and lethargic. Look for ulcers inside of his mouth, such as on his tongue and hard palate (roof of his mouth). If he has them the ulcers could be caused by Calici virus, which is an upper respiratory virus that can cause ulcers and joint pain as well.
Some cats get sicker than others and young, healthy adult cats seem to tolerate a respiratory infection and fight them off better than very young kittens or older cats. An upper respiratory infection in cats is just like a cold in you and I, and these are commonly caused by a virus. These are easily caught by breathing in virus particles in the air from a sneeze or nasal or eye discharge.
You can help him feel better by adding warm water to his food to make it smell more (they don't eat if they cannot smell) as well as making it easier to chew and swallow.
Also the more fluids he gets the better. Offer tuna juice, low salt chicken broth, run the tap if he likes to drink out of the sink. With the amount of drooling he is doing dehydration is a real concern.
Take him into the bathroom with you if you run a hot bath or shower as the steam will soothe his sore throat and airways.
If her nose become very congested you can use sterile saline to loosen the thick mucous and remove it. She won't like it but it will help him breathe and be able to smell his food better. You can also use sterile saline to remove eye mucous if it accumulates.
You can use Chlorhexidene oral rinses (like CET rinse) on the ulcers to keep them from getting infected by bacteria secondarily and keep them from becoming crusty and more uncomfortable.
Some lethargy is understandable, let him rest as he needs rest to get better. If your cats normally go outdoors keep him (them) inside until they are back to their normal playful selves.
If he runs a high fever (more than 104F), has a green or yellow nasal or eye discharge, stops eating even with coaxing and clearing his nose and eyes, or starts coughing or having difficulty breathing then he needs a veterinary exam. Not an emergency as long as he is eating but soon if he isn't eating or has a green or yellow nasal or eye discharge.
If his extreme drooling continues then he probably needs fluid therapy and pain medication from his veterinarian.
Sometimes these upper airway infections turn into pneumonia so that's what we need to guard against. In most cases antibiotics aren't needed and can contribute to a decrease in appetite so I don't tend to prescribe them unless I feel there is evidence of a secondary bacterial component. These include a green or yellow eye or nasal discharge, evidence of pneumonia upon listening to their lungs or an infection that lingers beyond the normal 7 to 10 days.
I highly recommend testing him for feline leukemia and feline immunodeficiency viruses if his respiratory infection lingers. These immunosuppressive viruses will make a simple infection much worse as they stop the immune system from fighting infections the way it was designed to do. If he isn't much better in 7 to 10 days he needs a veterinary visit.
Here is a link if you want to read more about Calici virus: http://www.pet360.com/cat/health/calicivirus-in-cats/42gJj403b0idm8v6BjfOiA
If he isn't improving with supportive care then he should see his veterinarian.
Let me know if you have any further questions.