Your questions seem to be squarely focused on chapter 13 of Lyddie. Lyddie has been at the factory long enough where she is familiar with the rules of the factory and the procedures of working a loom. In fact, Lyddie is proud of herself and her work at the factory. She is up to working 4 looms at a time. She is well on her way to achieving the financial independence that she so desperately wanted at the beginning of the book.
Lyddie does not want to sign the petition, because she knows that she could be fired and blacklisted from all of the other mills if she signs the petition. Lyddie needs the job and likes the paycheck.
The mill owners are focused on profits; therefore, worker safety is not their primary concern. The company sped up the looms, and for awhile Lyddie was okay. But the speed of everything, combined with long hours began taking their toll on Lyddie and other workers. Lyddie was eventually injured while working the looms.
Fortunately for Lyddie, Diana arranged a free visit to a doctor and cared for her through the rest of the recovery process.