A few comments to share,
1. At this age, environmental allergies, would definitely be less likely. If over a year of age, I would be saying the exact opposite, with environmental issues being far more common than food allergies.
To rule-out food, I strongly discourage ANY over-the-counter food. To do a food allergy diet trial correctly, you really want a prescription limited ingredient hydrolyzed food that you feed exclusively for at least 2 months. It may not take 2 months to see improvement, but you have to give it at least 2 months.
I like Royal Canin Hydrolyzed HP formula, or I have seen some specialists use a vegetarian prescription diet from Purina.
Treating for mange as a precaution is a great idea. Typically, if worried about scabies, a vet will recommend topical Revolution every few weeks for a few doses, or ivermectin injections every few weeks.
If getting raw, and the vet has done a skin scraping or is very confident that mange is not an issue, than doing a temporary steroid for comfort would be okay (if your vet agrees).
Hope that information helps