Your doctor should be ordering FT3 and FT4 ever time you have labs. TSH is totally inadequate by itself once a thyroid condition or disease has been diagnosed.
With Hashi's, many of us get a feeling of tightness or fullness in the neck. Do you think that's what you're experiencing? Also, hoarseness is common. If the thyroid is inflamed and enlarged, it can press on nerves to the vocal cords, causing changes in voice.
Definitely, speak to your doctor and insist he test FREE T3 and FREE T4. Some doctors believe that even though your antibodies are elevated, you should wait until labs are out of range before treating, others lean toward treating much earlier. So, a second opinion can be really valuable.
Anemia can be detected through a blood test. Your throat symptoms sound a little like silent reflux. Have you run your period symptoms by your gynocologist? Whether these are related to your thyroid or not, you can address the questions you have by calling and talking with your doctors' nurses and asking how to proceed.