Patients with a cortisol deficiency tend to wake up feeling well and deteriorate during the day. So even though your morning serum cortisol (total cortisol) is lower than optimal, the first thing I would do is see if you can get tested for free cortisol and also DHEA-S. And if your free cortisol is somewhat low, you could consider slowly raising thyroid levels in order to increase free cortisol. As thyroid levels come up, corticosteroid binding globulin (CBG) goes down, leaving more free cortisol. But increasing thyroid dosage must be done very slowly to allow the body time to adjust. Do you think you can get tested for free cortisol and DHEA-S? If you want to know more about DHEA, try this link.
http://www.hormonerestoration.com/DHEA.html
Cortisol should be highest at around 7-8 a.m.. Yet your morning serum cortisol was only 10.17, with a range of 6.02 - 18.4. Since morning cortisol should be in the upper end of its range, your level is too low. I have seen this same pattern with someone I know. Low cortisol is hard to diagnose, but having body aches and morning cortisol results below mid-range was indicative of inadequate cortisol levels. When started on hydrocortisone tablets taken at 3 different times of day, the person's body aches disappeared. So I think that is something you need to follow up with your doctor, to see about a trial of hydrocortisone.
If body aches are the only symptom bothering you at present, in view of your thyroid test results, I don't know that I would change anything there. Why add in a new variable? I would pursue the cortisol issue first. By the way if you do a trial of hydrocortisone, make sure you also test and supplement with DHEA-S to optimize that as well.
Before discussing further, what time was your blood drawn for the morning serum cortisol test?
I would like to mention one other possibility for causing body aches. Low cortisol is well known to cause such aches. Have you been tested for cortisol levels? What is your current thyroid med and the daily dosage?
Vitamin D is "not" the basic test for Hashimoto's... sharanbr, you've had all the antibody tests and you're getting the FT4 and FT3 tests, so you're good with testing...
Your thyroid levels look good and I agree that your vitamin D could be higher, but I'm wondering if you've been tested for other conditions, such as Rheumatoid Arthritis (or if you have plain arthritis), Lupus or other conditions, for which body pain is a noted symptom.
While vitamin D deficiency is common with those of us who are have hypothyroidism, getting vitamin D levels optimal is not a "cure all" for hypothyroidism or for Hashimoto's... There are many conditions that have the same or similar symptoms.
To answer your questions more specifically...
1) Thyroid hormone medication addresses only thyroid symptoms; as I noted, those same symptoms can be present with other conditions...
2) "If" the body aches are caused by hypothyroidism, the thyroid medication should alleviate the pain, "if" the medication dosage is optimal...
3) Hashimoto's "is" autoimmune and people "do" have their thyroid removed, sometimes because of it. Antibodies will, typically, go into remission following thyroid removal, because without a thyroid there's nothing to attack. It should be noted that TPOab, which is one of the antibodies used to diagnose Hashimoto's can also be present, in small amounts, with other autoimmune conditions...
For Hashi and other autoimmune conditions the basic test to be done is the Vitamin D3 blood test. In most the value is either deficient or not in optimal range.
Also levothyroxine and other thyroid replacement pills dosage should be adjusted after bringing vitamin D3 to optimal range as TSH value can decrease after supplementing with Vitamin D3 and thus there is a high chance that less amount of thyroid hormone pills needed.