It is possible that some of her symptoms are from the growth messing with her thyroid hormones. Joint pain, weakness, even the headaches can be symptoms of imbalanced thyroid hormones. The pain the the neck could be joint pain. I don't think it's because the cancer has spread to those areas. The hoarseness in the voice can happen when the growth is pressing a little on the voice box, since the thyroid gland basically wraps around it. Some people can have trouble swallowing, as well. It's not because it's spread to the voice box, it's only putting some pressure on it that should go away after surgery.
If the insurance is moving things slowly, there may not be much you can do. If you could get her to a cancer specialist that has experience in treating thyroid cancer or an Ear, Nose, and Throat doctor that has experience with thyroid cancer, that would be your best bet.
Since the growth is causing issues, regardless of whether it is cancer or not, they will probably want to take it out along with the half of the thyroid that it is on. Without a fine needle biopsy (FNA) or taking it out surgically and examining it, there is no real way to tell which type of thyroid cancer it is (if it is cancer at all, which it might not be--even benign growths can cause issues and need to be removed). What makes you specifically say it is follicular? Is it because of the joint issues?
If she does have thyroid cancer, the doctors will want to remove the entire thyroid and then most likely treat her with what is called radioactive iodine. Radioactive iodine is nothing like regular radiation for other types of cancer. It is a pill that specifically attacks thyroid cells and cells that came from thyroid cells (ie thyroid cancer cells) where ever they may be in the body. The cure rate isn't 100%, but for most it is very high, depending on age and how advanced the case is. But from what I've read, the more aggressive the treatment, the better off the patient is.
I wish I could offer you more specifics. I would encourage you to keep reading up on this, but not get too attached to one specific diagnosis. Write down what ever questions you and your mom have and take them to the doctor visits and make sure they are answered in a way that you both understand. Make sure you and your mom know what options there are and what the next steps are in tests and treatment.
Let us know how things go. I know it seems like a slow process, but with thyroid growths, it often it ends up being a wait-and-see game for everyone, regardless of insurance. Feel free to re-post if you have more questions. Best of luck to you and your mom.