T4 is Total T4 and that's considered to be obsolete, as is the T3 Uptake test. You should ask for the Free T3 and Free T4, along with TSH and the 2 antibody tests, as mentioned by Red_Star.
Your daughter's symptoms, including the high cholesterol and triglycerides certainly sound hypo.
Yes, that sounds thyroid related. Low T3 uptake is an old outdated test which has been replaced by T4. In short the doctor did two tests to tell you the same thing - T4 is elevated. :) The tests I requested when suspecting Hashimoto's thyroiditis - TSH, free T4, free T3, thyroid peroxidase antibodies (TPOAb), thyroglobulin antibodies (TgAb).