The hands-down, most common reason for dogs licking paws or chewing their paws is skin disease. Anything that causes the skin to become itchy and inflamed will generally do the trick, and the paws are an easy target for your dog to lick.
If your dog has always been, for lack of better words, a ‘licky dog’ then it may just be a normal behavior for your dog. If it is a new behavior, there can be underlying medical or behavioral issues to address.
If your dog is coughing, has a runny nose, or is slightly off-colour, you may be inclined to think they have a cold. However, the human cold is not contagious to dogs. So the short answer is no, your dog doesn't have a cold.
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