Hi there, sorry you're going through this, it can be very stressfull.
I don't have any answers for you, but my son has some similar symptoms so thought i would share our story so far...
He is 15 months and has had severe eczema since 3 months old, we have had it under control now since about 12 months with only occasional mild flare ups.
He has also had enlarged lymph nodes on his head, neck and body since about 4-5 months, most of these they believe were due to the eczema, but he has a very prominent one on his neck (approx. 1.5cm+ that can be seen at all times) that has never gone way even with antibiotics and no signs of other infection.
He is also low weight at 7.6kg at last check with no weight gained in the last 3months, but our paed is taking a 'wait and see' approach with everything (we are currently in the process of changing paeds :/)
He also has had very blue hands and feet on occasion and was diagnosed with Raynauds phenomenon, although again, no more investigation was done, it was just 'labeled' ...i have noticed lately that they don't turn as blue as often anymore. Maybe your DD has similar? It may or may not be related to her other symptoms?
Another thing could be allergies, is your DD allergic to anything that you're aware of?
This is probably not much help to you, but you wanted to let you know you are not alone and we are currently investigating similar things with our DS...i hope you get some answers soon