I am answering this almost two years after it was asked, but I hope an answer can still be useful!
You are correct that Hashimoto’s disease (also known as Hashimoto’s thyroiditis) is the most common cause of hypothyroidism in those aged over six. In younger children hypothyroidism is most often congenital. So yes, you almost certainly have hypothyroidism due to Hashimoto’s disease.
Hashimoto’s thyroiditis is an autoimmune disease. This means that something triggers the immune system to produce antibodies that cross-react with normal parts of the body. People with Hashimoto’s disease can have antibodies in the blood to various components of the thyroid gland. The most common are anti-thyroid peroxidase (anti-TPO) antibodies. You may have had a blood test for this and you could certainly ask your family doctor or endocrinologist. Either way, the treatment is similar to other causes; namely replacement of thyroid hormones with levothyroxine.
Autoimmune disease in general
The origins of autoimmune disease seem to be a complex mix of genetic and environmental factors and possibly pathogens. Much autoimmune disease is due to various forms of immune hypersensitivity. Specifically, Hashimoto’s thyroiditis is an example of type 4 hypersensitivity (along with coeliac disease and multiple sclerosis), while type 1 causes most allergies and anaphylaxis.
A good example is autoimmune disease after infection by Group A streptococcus bacteria (a common cause of throat infections, skin infections, scarlet fever etc). In some people, the antibodies that the body makes against streptococcus will cross-react with other body tissues and damage them, causing a number of conditions:
See this paper for more information about autoimmunity after streptococcal infection. This is an example of type 3 hypersensitivity.
Various autoimmune diseases are associated with one another. Having one can make another more likely. For example, there is a link between type 1 diabetes and autoimmune hypothyroidism. You mention that you may have lupus, which is also autoimmune.
You also mention fibromyalgia. Currently, fibromyalgia is not thought to be an autoimmune condition. Proposals of an autoimmune cause have not yielded evidence of this in research. It is likely to result from a complex interplay of internal and external factors with neurochemisty and neurophysiological processes. There is some overlap in symptoms, but this is not surprising as pain, stiffness, fatigue and low energy are some of the most common symptoms there are and can have many causes.
I see from another question of yours that you had a bad experience with a general practitioner taking the very outdated view that fibromyalgia is not a real condition! It most certainly is real, even if not well understood yet, and can cause a lot of suffering. Whatever the underlying cause turns out to be (there may be several), it does not have to have an autoimmune origin to be real! :)