I have Hashimoto's and just found out that I am anemic. I was tested for anemia 6 months ago and I wasn't then. Was diagnosed with Hash's about 2 months ago. I wondered too if the two were related.
In my opinion the answer is no. From data such as Fig 6-7, in this link, you can see that for an individual, the correlation of TSH on FT4 is very high. This would leave no room for any other variable to significantly affect the TSH level.
As for your other question, I think it depends on what you mean by thyroid disorders. Most of the hypothyroidism that we talk about every day on the Forum can be traced to a specific cause like Hashimoto's, or the aftermath of a TT, or RAI. As such, it would be hard to argue that mineral or iron deficiencies caused the thyroid problem. I have seen study data that reported that the conversion of T4 to T3 is adversely affected by low levels of iron, zinc, and selenium. As such, those deficiencies would contribute to the degree of hypothyroidism, by depressing the levels of FT3.