What are some historical facts from the book Lyddie that don't relate to the labor movement, women's rights, slavery, or child labor laws?

This question is hard.  Really hard.  The reason is because the book mainly focuses on the items that you have listed in the question.  I believe that I can provide you with two pieces of historical evidence that the novel mentions that do not deal with the topics mentioned in the question.  

The first historical fact is the emergence of colleges being open to women, and women seeking to further their education.  The novel ends with Lyddie deciding to go to college after being fired, which absolutely would have been possible for her.  Lyddie mentions Oberlin College by name.  The college was founded in 1833 and began regularly accepting female students in 1837.  It was the first college to do so.  

A second historical fact that the novel makes in a very brief passing is the fact that tuberculosis was quite common in the textile mills.  When Diana shows Lyddie how to pull the weft thread through the shuttle with her breath, she calls it the "Kiss of Death."  At the time, nobody knew it was tuberculosis or that it could be spread through saliva.  But Diana's reference does point out that it was common knowledge that the disease was more common in the textile factories than in other places.