I think you're right to get a second opinion... TSI results aren't just a raw number, such as your 394...TSI is a percentage. If the result was 394%, then I'd say yes, you probably have Graves Disease and it's entirely possible you could have; however, I'd certainly want thyroid levels tested again and not just the TSI.
Since you were on Levothyroxine for 8 years, we'd assume that you were hypothyroid, so we'll start there... were you ever tested for antibodies to determine what caused you to be hypo in the first place? Hashimoto's is the # 1 cause of hypothyroidism in the developed world. There are 2 antibody tests that determine Hashimoto's. Those are Thyroid Peroxidase Antibodies (TPOab) and Thyroglobulin Antibodies (TgAb). You need to have them both, because some of us have one or the other, some have both...
Some people, actually, have, both, Hashimoto's and Graves at the same time, with one or the other being dominant at any given time.
All of that said, just because your TSH went down to 0.02, does not mean you were even hyper, unless your actual thyroid hormone levels were too high and you had symptoms of being over medicated... You said your T4 and T3 were within the normal range... Were those Free T4 and Free T3? They aren't the same as Total T4 and Total T3 and don't provide the same information... If you have those lab results, it would be great if you could post them, here so we could see them. Also, be sure to include reference ranges, as those vary from lab to lab and have to come from your own report.
Also, please tell us what your symptoms are, as some symptoms can apply to, both, hyper and hypo...
Your TPOab (Thyroid Peroxidase Antibodies) is positive for Hashimoto's and your TSI is positive for Graves Disease, indicating that you have both.
When were these labs done? The Free T3 and Free T4, both indicate hypothyroidism, not hyper... Likewise, most of your symptoms indicate hypo, as well.
The only symptoms that, typically, go with being hyper are the heart palpitations, racing heart rate, sweating and frequent bm's, but many of us have those when hypo, as well. I had them all, except the bm's, when I was terribly hypo. I even had/have very low TSH and have had to fight doctors for years because they insist I'm hyper, when I'm not.
Are you currently, on any thyroid medication at all?
Typically, we recommend that Free T4 be about mid range and Free T3 to be in the upper half to upper third of its range... that's where most of us find that we have the least hypo symptoms; both your Free T4 and Free T3 are only at 20% of their respective ranges, which are far below the recommended levels.
I'd say you do have Graves Disease, because one doesn't have TSI as high as yours unless they do have Graves, but that doesn't mean you're are or were hyper... You are only hyper if your actual thyroid hormone levels are high in the range or over range and yours most certainly aren't, not to mention that most of your symptoms are those of being hypo, not hyper.
When you have, both, Hashimoto's and Graves, either one can be dominant and it appears that for you, Hashimoto's is dominant.
Soy is a goitrogen and goitrogens can block/inhibit production/absorption of thyroid hormones. And yes, the fact that you just had a baby could have had something to do with your hormones being out of kilter. You should ask to be tested again, and make sure they test Free T4 and Free T3, not just TSH.
It should be remembered that TSH is only a messenger hormone and does not always correlate with actual thyroid hormones. This is a fact that many doctors seem to forget quite often. It's also important to note that many endocrinologists specialize in diabetes and are not good thyroid doctors.
The little red dots on your thighs are most likely, what's called Petechiae. They're just little red blood spots, caused from bleeding under the skin. They can be caused by a number of things, including prolonged straining (such as that from vomiting, crying coughing, childbirth), certain medications, some infectious diseases, other medical conditions, diseases, injuries, etc. Since you had a baby just before yours showed up, yours could be caused from that and may be nothing serious, but you should talk to your doctor to be sure.