I'm not sure how it works in america (sounds very confusing to me) but the NHS do blood tests for thyroid as a par for the course in women over 25 as it's a condition suffered by over 10% of the female population in some way (any form of thyroid condition that is). I have hypothyroid, as does my Mum, her sister, her mother, her mother before that....etc etc etc. It was kind of when not if with me lol. I would be asking for blood tests for T4 and TSH, asap if you're worried. Thyroid conditions can manifest themselves in all sorts of ways, so make a really detailed list of what you think is linked. For me it was: temperature fluctuations(continuously cold), funny taste in my mouth, hair falling out like crazy, skin really dry, nails really brittle, really low feeling (am actually now on anti-d's aswell as levothyroxine). I recognised a lot of it from my Mum and her experiences so I went to the Drs and within 3 days I'd got my results and some new medication.
As for teething - babies can be born with teeth already through. That's a really weird thing for a Dr to say. Both my boys started at 3 months. Earl didn't get any teeth until 8 months, and Eddy is yet to get any pearly whites.