Without being able to see your dog in person, it's pretty much impossible to tell you why he's looking the way he does or what could possibly be causing it. The list of things it 'could be' is nearly endless: allergies, infestation of parasitic mites (mange), thyroid problems (although he's awfully young for that!) or a bacterial/yeast infection are just a few. My suggestion is to call your regular vet in the morning and schedule an appointment for them to see your pup and see what they can determine. They'll most likely do a skin scraping of the affected areas (where they use the edge of a scalpel blade to gently scrape off the top layer of skin, looking for any mites), and possibly run some blood work to see if there is a thyroid issue. In the meantime, you can give a little liquid Benedryl at the dose of 1-2 mg per pound of body weight (meaning if your dog weighs 20 pounds, you can give 20 mg the first dose, and if that doesn't help, you can give 40 mg at the next dose) one time every 8-12 hours. Also, here are some "home remedies" that might help keep from itching and making it worse until you can get in to see your vet:
****Keep in mind to NEVER use any of these suggestions on broken skin!!****
*For some itches, we suggest soaking in a baking soda bath. Adding 1 cup of baking soda to a tub of warm water and then soaking 30-60 minutes. Take dog out and let them air dry (this works best if done outside or in an area where there isn't carpeting as wet dogs drip...a lot). Local itches can be covered with a paste of 3 parts baking soda to 1 part water.
*Oatmeal is an old favorite. Adding 1-2 cups of finely ground oatmeal to a warm bath sometimes can calm even the most irritating of itches. Just make sure the water isn't too hot, or you'll have breakfast for a week in your tub!
*Lots of folk remedies call for using lemon to help with itching. They contain anesthetic and anti-inflammatory properties that may help reduce itching. If nothing else, your dog will smell good!
*Splashing the affected skin with basil tea can help reduce itching as well. Basil contains large amounts of a substance called 'eugenol', which is a topical anesthetic. To make the tea, place 1/2 ounce of dried basil leaves in a 1 pint jar of boiling water. Keep it covered to prevent the escape of eugenol into the air. Allow to cool, then dip a clean cloth into the tea and apply to itchy skin as often as needed.
*Mint tea works about the same way, except the anti-itch property is menthol, not eugenol. To make the mint tea add 1 ounce dried mint leaves to 1 pint of boiling water. Cover and allow to cool. Strain the leaves out and then dip the clean cloth in the tea and apply to the itchy areas.
*If you're saving the basil for spaghetti sauce and the mint & lemons for a glass of lemonade, you can use thyme tea. Once again use 1/2 ounce dried thyme leaves and 1 pint of boiling water. Cover and cool. Strain and use clean cloth to apply to affected areas.
*Last but not least is the old standby, aloe vera. The same properties in aloe vera that help with burns also work like a dream on itches. Snap off a leaf, slice it down the center and rub the gel on the itch.
I hope this helps!!
I have seen cradle cap...and from your description of the odor and the appearance, my guess is that she has either mange or a bacterial or yeast infection.
Of course, it could be allergies, which could explain the vet wanting to give prednisone.
Perhaps you could talk to your vet about doing a skin scraping. That's where they use the edge of a scalpel blade to scrape off the top layer of the infected skin and look at it under a microscope to diagnose what she has going on.