D.B.
I think you are going about this the hard way. You're trying elimination diets, and that is a nightmare because a) not everything is labeled clearly and b) it takes weeks or months to get any results. Do you really want this to be the life you and your child lead?
Have you been listening to anything about epigenetics? Don't' go crazy with internet searches - just concentrate on major news outlets like Time and Newsweek, people you can be sure do decent research. You already say that issues run in the family - that is epigenetics at work. The huge rise in autoimmune conditions such as the ones you name (and there are others) has prompted the last 15 years of research into food science and what the effects are of various food peptides, isoflavones and so on.
I have worked with hundreds of people with significant food "allergies" and it's important that you realize how much the science has changed. The most heavily researched nutritional ingredient in the last decade has been shown to reduce inflammation, increase immunity, and repair the damage to the epigenome (what's on top of and around the genetic material in each cell and which causes individual genes to turn off or on - this switching is what tells the cell what function it's to perform, which varies tremendously according to the type of cell. Nerve cells do something very different from retina cells or bone cells or smooth muscle cells, right?) So the reason the autoimmune conditions run in both families is because of changes in the epigenome of many cells. This is a reversible condition.
You've seen the huge increase in allergies - a generation ago, every child took a peanut butter sandwich on wheat bread and bought a carton of milk at school. Kids had birthday parties with Mom's cupcakes. Hardly anyone had asthma. School nurses weren't going on field trips with 180 kids, 100 of whom have medications (a statistic from a field trip the other day near me). So the way we have dealt with these issues - from massive elimination diets to medications to endless testing - has not worked. At all. Parents are, understandably, panicked about what it can all mean.
In my experience, I would say your child's issues are all related to the same thing. In a few months, you could significantly reduce these reactions - maybe not be able to give her gluten, but I would estimate that a number of conditions could be significantly alleviated. A colleague of mine has a daughter with CD, and while she does not eat gluten, she is completely symptom-free when any cross contamination occurs. There is zero reaction if she's near someone who is eating gluten, she eats in restaurants, and so on. I think it's been 8 years with no problem. The reduction in inflammation and the repair to the epigenome could reverse all these other things, and the preventive effect is unbelievable.
With research and patents (very unusual on food products) and clinical studies, I think your daughter could totally turn around in a short time. The eczema cases that have cleared up are in the thousands, for example. You might want to look into this more for your connective tissue disease - I've met recently with quite a few people who are out of pain, out of a wheelchair, and more, so these are significant cases of impairment which have seen improvement. And it's so much easier to add the necessary nutrient than to remove dozens which may or may not be the problem.