I might add a thyroid ultra sound to the list of tests, to confirm or rule out thyroid nodules.
I agree with the above, but I'd also add a TSI (thyroid stimulating immunoglobulin) to the tests suggested. TSI will confirm or rule out Graves' Disease. Your symptoms almost sound like they could be hyperthyroid symptoms.
Make sure those tests are for Free T3 and Free T4, not Total T3 and Total T4. Also make sure the lab people know it is to be for the Frees not Totals, to prevent them making a mistake. Since Hashimoto's Thyroiditis is the most common cause of hypothyroidism, you should also check for thyroid antibodies, by testing for TPO ab and TG ab.
It's possible, but not certain. The only way to really know for sure is to get seen by a doctor. You may need to specifically ask to get your TSH, t4, and t3 levels tested. (Most doctors only to TSH) But because of an irregular heart beat they may also send you to a cardiologist too. Many things can cause weakness, fatigue, eating problems, and irregular heartbeats so you need to see a doctor, and it's possible it may be more than one thing.