Agree with gimel.
It is a thyroid issue. She needs an endocrinologist.
The TPO ab and TG ab test results indicate the presence of Hashimoto's Thyroiditis, which is the most common cause of diagnosed hypothyroidism. Hasn't the doctor mentioned this? With Hashi's, the autoimmune system erroneously identifies the thyroid gland as foreign to the body and produces antibodies to attack and destroy the gland. This takes place over what is usually an extended period.
Along the way, as the gland is gradually destroyed, there is a diminishing amount of thyroid hormone produced. This typically results in elevated TSH levels and diminishing levels of the biologically active thyroid hormones, Free T3 and Free T4 (not the same tests as Total T3 and T4), requiring gradually increasing dosage of thyroid med to relieve hypothyroid symptoms.
Hashi's can also result in nodules on the thyroid gland that tend to leak hormone faster than normal. This can result in cycles of hyper then hypo. Your daughter's low TSH may be a result of such nodules, or she may also have a pituitary issue that results in low TSH levels, even with low Free T3 and Free T4 levels. The presence of nodules can be determined with an ultrasound, which would be a good idea. At any rate, at this time all this is academic. The real issue is to get your daughter's doctor to understand what is going on and increase her meds gradually to a level that will relieve any hypo symptoms (please tell us about any of those). The dose of 25 mcg of Synthroid is way too little, based on the limited test results above.
To facilitate this, she needs to be tested for Free T3 and Free T4, along with the TSH. The above tests are for Total T3 and T4, which are somewhat outdated and not nearly as useful as the Frees. If the doctor resists, just insist on it and don't take no for an answer. Frequently hypo patients are also low in other important areas. I suggest that she should also be tested for Vitamin D, B12 and ferritin, along with a full iron test panel.
As for the uticaria, that can be associated with a number of causes, but in view of her Hashimoto's, I'd be suspicious of that as the cause. First, her being under medicated and still hypo can contribute to inflammation of the thyroid gland, in addition to the effect of the Hashi;'s. Also, we have had members previously report having what has been called Hashi's rash, from the effect of the antibodies. Getting her meds increased should help tamp down the level of antibodies and also reduce the load on the remaining thyroid gland tissue. Also, there are scientific studies that have shown that selenium helps reduce the effects of Hashi's. Note that I said helps, not cures. Selenium supplements can help, but overdosing can be toxic, so test first and then supplement carefully.
So, in summary she needs some additional tests outlined above, an ultrasound of her thyroid gland, and she needs her thyroid meds increased to relieve symptoms. When additional test results are available, please get a copy of the lab report and post results and their reference ranges and members will be glad to help interpret and advise further.