Losing the outer part of the eyebrows is a classic symptom of hypothyroidism, as is weight gain/inability to lose, cold intolerance and fatigue.
While your TSH is "normal", your doctor did a great disservice by refusing to order Free T4 and Free T3 (not just T4 and T3, which indicates Total T4 and Total T3, which aren't the same tests). It's very possible you could have a condition called Secondary hypothyroidism, in which the thyroid actually works fine, but the pituitary isn't producing enough TSH to stimulate it adequately to produce the T4 and T3. Secondary hypothyroidism is a pituitary issue, not a thyroid issue.
You say you're really a very hairy person... if you have a pituitary issue, there could be other hormones affected, as well. You should talk to your doctor and ask to get reproductive hormones tested.
There's a condition called PCOS (Polycystic Ovary Syndrome) that some young women get that also cause weight issues, excess hair growth, etc. PCOS can cause conception and other problems if not treated.
We can tell you where you can get tests done without a doctor's order, but insurance won't cover them and you may need more than just the thyroid tests. If your current doctor won't help you, don't hesitate to find another...