Please post the reference ranges for the results you've posted. Reference ranges vary from lab to lab so have to come from your own report for the best comparisons.
Are hair loss and the menstrual issue the only symptoms you had prior to starting on the birth control?
Thank you for providing the reference ranges. Although your TSH looks wonderful and, unfortunately, that's what most doctors are going to look at, your actual thyroid hormones are much lower than is recommended or where most of us feel comfortable. I would certainly consider you to be hypothyroid with levels like that.
Your TgAb could indicate Hashimoto's, however, we would expect TSH to be higher with such low hormone levels, though. Your body is converting more Free T4 to rT3 than to Free T3 for some reason.
Since your TSH is as low as it is, you could also have a condition called Secondary or Central hypothyroidism. This is when the thyroid actually works fine, but there's a problem with the hypothalamus/pituitary axis.
One thing I do see is that your Ferritin is too low. Ferritin is the iron storage hormone and iron is necessary for the metabolism of thyroid hormones.
Although iodine is the main component in producing thyroid hormones, it requires iron to be fully utilized. According to nahypothyroidism.org, research shows that iron also plays the following roles:
"Iron (in the heme form) is necessary for thyroid hormone production.
Iron is necessary to convert T4 to T3; T3 is the active form.
Iron status (ferritin) inversely correlates with TSH.
Iron is critical for thyroid peroxidase activity (TPO)."
There are other vitamins/minerals necessary for the production/utilization of thyroid hormones, as well, such as selenium, zinc, vitamin D, etc.
Do you have a copy of the ultrasound report? Does it have an impression printed at the bottom or recommend any follow-up? Sometimes, the report will mention that the image is consistent with Hashimoto's...