Yes I was both hyper and hypo from leakage symptoms from Hashimoto's thyroiditis and also was very hyperthyroid for two months due to Graves antibodies. They call this Hashitoxicosis. This was before I started treatment.
Yes, it's possible. Have you ever had antibodies tested to determine whether or not you have Hashimoto's Thyroiditis? Hashimoto's is an autoimmune condition in which the body, for some reason, determines that the thyroid is foreign and produces antibodies to destroy it.
It's quite common for one to be hyper in early stages, as the thyroid spurts hormones; in addition, we often have nodules that produce hormones independently of the thyroid that can cause hyperthyroidism. As the disease progressively destroys the thyroid, it produces less and less of its hormones and we eventually become hypo.
Rarely, one can have, both, Graves and Hashimoto's at the same time.
Although it's not unheard of to go through menopause at age 37, it's possible that you aren't really in menopause, as being hypo definitely affects the menstrual cycle.
Please post the reference range for the Free T4. Ranges vary from lab to lab and have to come from your own report. That said, if you have at TSH of 15 and Free T4 of 0.6, you certainly need your dose of Levo adjusted. Your TSH is way too high and depending on the reference range for the Free T4, your FT4 is, most likely, way too low.
Aside from that, has no one ever tested Free T3? That's the hormone that's used by nearly every cell in your body and correlates best with symptoms. If you've ever had it tested, please post the result(s), with reference ranges.
The symptoms and labs you've noted indicate that you may still be severely hypo. If your doctor won't increase your dosage, find a different doctor asap... Levothyroxine is dosed in mcg as opposed to mg; is it possible you're on 50 mcg Levo?
To test for Hashimoto's, ask for Thyroid Peroxidase Antibodies (TPOab) and Thyroglobulin Antibodies (TgAb). You need them both, as some of us have one or the other and some of us have them both.