Lick deterrents - These are bad for a number of reasons. Your dog already knows you don't like the constant chewing and licking. If they could stop they would. The main problem with lick deterrents is that they are alcohol based. It stings like crazy when you spray it on raw or inflamed skin. It also irritates their sense of smell. But it won't work regardless. A dog will lick anything off it's fur, especially if it tastes bad. They can't help it.
Vinegar - Although it has a well-deserved reputation as a germ killer, as well as making hair shiny, vinegar is really bad for your dog's skin. It's a strong acid that burns away the germs, and healthy skin along with them. It extremely irritating to a dog's sense of smell and if you get some in their eye it stings like crazy. If they have any raw or damaged skin, it's going to burn there too. Bad. It's just not a good idea to have vinegar around a dog at all.
Tea Tree Oil - Although it's relatively safe when diluted, make sure you never use tea tree oil undiluted. It's basically an insecticide/bactericide manufactured by the tree to protect itself. In undiluted form it can burn skin or cause an overdose. Products that have less than 1% tea tree oil are not bad, but the smell is really awful to a dog. They take care of bacteria, but that's not really the problem. The real problem is too much histamine causing inflammation in the skin cells and sending itch signals to the brain. Tea tree oil and it's relatives are very good at killing germs but itching and inflammation isn't caused by germs and the smell is way too intense for dogs.
The same is true for other aromatic oils that are commonly used to prevent itching - these include peppermint, menthol, eucalyptus, clove, camphor, all of them are just too intense for a dog's nose. The other reason you don't want to use serious aromatics on a dog is because they are too strong to put on raw skin. They will sting like crazy if your dog has broken the skin or has sores or cracks. The anti-itch benefits of these ingredients aren't meant for raw skin or damaged skin on your dog's paws or anywhere else.
Benzocaine - This topical anesthetic is found in teething gel and sunburn lotion. It's also found in dog paw spray. True, it will numb your dog's skin, but only for about twenty minutes. After that, the effect goes away rapidly. The problem is that benzocaine restricts blood flow to the area and starves skin of oxygen. It's similar to what happens to people's noses when they snort cocaine. It isn't good. The skin is left much worse off than before for only a few minutes of relief.
Hydrocortisone cream - The problem with this is that your dog will lick it off. When ingested hydrocortisone can cause bleeding in the intestines. The result is foamy stools with streaks of blood.
Antifungal cream - This could be anything that ends with zole. Fluconazole and Clotrimazole are two common creams for yeast (fungus). They may kill some fungus, but the yeast or fungus is a secondary infection that would probably clear up if you can restore the skin's integrity. Unfortunately your dog will lick all of it off his skin and fur. If you have to clear up a fungal infection, it's best to go with oral medication from the start.
Medicated shampoo - The only one that really works is 10% hydrocortisone. It's good for severe flareups but requires a 20-minute soak and isn't really going to last more than a few days. It's safe because there's no drug left on your dog once you've rinsed it off. The downside is that it washes away all the natural oils that keep your dog's skin healthy. If you use this, use it sparingly.
All other medicated shampoos are either for the wrong thing, or they are way too harsh for dogs.
The best shampoo is a puppy shampoo, made with coconut cleansers, oatmeal and aloe vera. There are several good ones out there. Just make sure it doesn't have any other ingredients. Try not to bathe your dog more than absolutely necessary. Even the mildest shampoo washes away natural oils and dries out their skin.