No...not too young for the scratch test. My son had his first scratch test at 10 months. It is actually considered to be a more sensitive test.
I noticed that someone mentioned that most food allergies don't show up until age 2...that is old data. Nowadays most children seem to be born with food allergies and the onset can even be seen in infants who are being nursed and are allergic to what the Mother is eating (my son was so allergic he would throw my milk up even in the NICU...they told me he was too young but when I removed dairy he stopped throwing up and at 10 months presented with a level 5 allergy to dairy).
That being said, however, there are some problems with food allergy testing in general. The main one is that the samples used in these tests are heavily diluted and don't always show that an allergy is present.
The new gold-standard is a food allergy challenge which should be done IN the office. However, 15 months is too young for a challenge since your daughter would be unable to assist verbally in letting anyone know a reaction was imminent and many doctors don't offer to do this in the office.
We see Jon Van Wagoner (http://www.swallergy.com/index.htm) and have since my son was 10 months old. We have been very happy with his care. Find someone that you are comfortable with.
As far as her reactions go the dairy reaction COULD (and I'm not saying it is, just one more possible piece to the puzzle)be more of an intolerance or a lactose thing. Her egg reactions sound very much like an allergy, though.
Also, there are different types of allergy testing. Did your pediatrician order the tests or did an allergist? Some of the tests are less reliable then others.
As far as whether or not the testing could be wrong... absolutely. If you can see her reaction it doesn't really matter what the test says. The proof is in the pudding, so to speak.
If it were my child I would remove all dairy and egg products from her diet and see if her symptoms improve. And I would make an appt with an allergist. By the way, children who are allergic to dairy should also avoid goat's milk.
As a Mother of a HIGHLY dairy allergic child I can tell you it really does get much easier. Hang in there! If I can help please feel free to email me.