Allergist update:
No new allergy tests.  According to the allergist we met with, it seems unlikely that we will find a cause at this age other than the top allergens that she has already been tested for.  In my exhaustion, I think I have to agree- at some level, she could be allergic to everything and anything.  And even if she is, that particular thing may not even be related to her eczema.  We are going to reintroduce (carefully) the foods that we have eliminated, and keep cleaning to get rid of dog dander.  After three weeks of the elimination diet with no change, the foods seem unrelated to the eczema.  Patrick and I have to really step up our skin care for her, with tepid baths once a day and major applications of cortisone and aquaphor twice a day to get her skin under control.  This seems like a lot of work, to be sure, but far less work (and anxiety) than trying to keep her from exposure to most foods. 
I had just joked to Patrick yesterday that it seems like all women want is a man to tell them what to do (for anyone who has a hard time picking up sarcasm in type, this is it).  Yet I sat there today in a doctor's office, desparate for this man to just tell me what to do.  In this situation I am absolutely worn out by trying to understand what is going on and decide what the best course of action is.  So I will listen and follow directions completely and embrace with blind faith the idea that THIS will help my baby.  And it better.