Yes. I recommend that you take the baby off of all of the foods and re-introduce then one at a time per day. This will tell you which food or foods the child might be allergic to.
Yes it can. Anything in which the skin may react to (including many foods) can cause a rash (and in some cases, quit severe). If this has happened, I would recommned taking your baby to your family physician to run some tests. Hopefully you know the exact food the allergy is.
Absolutely, although it's probably unlikely to be the only symptom of a food allergy. There are lots of things that can cause a face rash, from infant acne to eczema. I would contact the pediatrician, especially if there are other food allergy symptoms.