Yes you can take Synthroid w/Armour. Not many do - but it is done.
I don't think you are getting the actualy free testing.
If you could post the reference ranges for your labs that might help. The most important lab to look at is T3, specifically Free T3. This is the usuable amount of thyroid that you have so if it's low it could definitely be a sign that you need thyroid medication especially if you are having symptoms. Hair loss also can be a sign of hyperthyroidism so you want to judge by all your symptoms to see where to start.