You take him/her to a well puppy vet appointment and the vet will be able to give you a good idea based on its teeth. Here is an article that talks about how the vet makes an educated guess: http://www.dogster.com/lifestyle/how-can-you-tell-how-old-a-dog-puppy-is-age-determine-ask-vet
But really – every puppy should see a vet within a few days (3 at most) of arriving at your home. There are shots the puppy needs to stay healthy, the vet can give you an idea of age, answer questions about diet, tell you if the puppy is healthy or if you have problems brewing. So set up the appointment now.
Congratulations on your new family member!
I would take her to a vet for evaluation – sounds like she has an allergy or skin condition developing and you will need help to resolve it.
In the mean time I would remove all grains from her diet, including treats. Dogs don’t need grains and don’t digest them well. And while not every dog is allergic to grains the can cause just enough impact on the immune system that other allergies can get a food-hold. Go to a good pet store (not grocery store, big box store, nor a pet store that sells puppies) and ask to be shown to the grain-free dog food area. Then start reading labels, you want a dog food that has real meat listed as the first 2 (or even better, 3) ingredients. Real meat is easy to spot, it will say lamb, elk, bison, turkey, salmon meal, whitefish meal, etc. (meal is the entire animal ground up and dried).
Don’t buy any food that lists ‘by products’ in the ingredients – you would be paying premium price for an inferior product.
Avoid chicken and beef – many dogs are reactive to these sources of protein.
If the vet recommends Benadryl use Zyrtec instead – I find that it works better for dogs.
Good luck