He has been licking his paws excessively and throwing up water, at least three time in 12 hours. Is it an emergency?

Because of the vomiting, I recommend getting your pet checked by your vet as soon as possible. Continued vomiting is very concerning and can lead to dehydration and electrolyte imbalance as well as low blood sugar. He might have eaten something that either disagreed with him or something that has gotten stuck somewhere in his gastrointestinal tract. He could be sore from a hidden injury you are unaware of, he could have been exposed to some sort of toxin or irritant in his environment or this could be early signs of another condition or disease process. Your pet may need IV fluids to help support his/her system and to be fasted at the same time in order to give the gut a rest. Your vet can do an exam, blood work and possibly x-rays to determine the cause of the illness and suggest an effective treatment plan.

You can give your dog a bath, however if your dog is not eating and is dehydrated, he should probably see a vet. Dehydration can quickly escalate into a severe issue. Animals can go downhill quickly and simple dehydration can quickly turn into organ damage and severe illness.

I recommend getting your pet in to be seen by your veterinarian right away. Continued vomiting and weight loss together are very concerning and can lead to dehydration and electrolyte imbalance as well as low blood sugar. He might have eaten something that either disagreed with him or something that has gotten stuck somewhere in his gastrointestinal tract. He could be sore from a hidden injury you are unaware of, he could have been exposed to some sort of toxin or irritant in his environment or this could be early signs of another condition or disease process. Your pet may need IV fluids to help support his/her system and to be fasted at the same time in order to give the gut a rest. Your vet can do an exam, blood work and possibly x-rays to determine the cause of the illness and suggest an effective treatment plan.

there is a chance that it's just him eating too fast then drinking too much, otherwise it could be an upset stomach (usually an infection or indigestion) or something stuck in his stomach (a toy or other foreign body). If it carries on put him on boiled boneless chicken and rice for a couple days, then if it still persists take him to the vet to receive adequate treatment.