http://www.totaldiscountvitamins.com/product/26026/thorne_research
This mineral supp is a fairly good one, and one capsule has 50 mcg of selenium and a fairly reasonable dosage of other stuff for a kid. Thorne also has stand-alone selenium supps as well--you'd need to decide how much you think is necessary/safe (too much over a long time period is toxic, like most minerals), I think 50 or 100 mcg is still in the reasonable range. If your DD doesn't already swallow pills, starting with smaller capsules that you coat with something tasty like maple syrup can help with the learning process. Once they learn, at least for my kids, moving up to adult-size pills was a pretty easy process, or for stuff that I need to open the capsule for, hiding it in between 2 layers of applesauce and just taking it as 1 bite is another option.
I like Thorne because it has real folate instead of folic acid, it's important for some people--not an issue just with the multi-mineral above, but if you want a vitamin supplement (from wherever), it IS something to consider.
re: gluten, how quick someone responds is an individual thing. I felt better in a day and a half but my son took 3 months to show any change. I'd think you'd want to keep it out for a while, get to a steady, stable dose of thyroid supp, and then see if adding it back in changes her response--or see if other stuff has changed in the meantime. For my DD, weird stuff changed, she stopped wetting the bed at night and she started napping again, stuff like that, plus digestive improvements which seems more normal for a food.
Have you considered GF for the entire family, or is there a reason you don't want to do that? Because oftentimes, it seems like people in a family have related issues, and similar solutions help even if the individual manifestations of the problems are different--all four of us have gluten issues, but different ways of showing it, and DH is the slowest to show improvement. But if you're considering bloodwork for celiac for another family member, you _do_ want to stay eating a fair amount of gluten til the test.