Cholesterol usually rises to protect the body from various acids, toxins, and infections; eg: trans fatty acids, mercury, systematic fungal infections, H pylori bacteria.
Thyroid hormone has multiple effects on the regulation of lipid production, absorption, and metabolism. Low levels of thyroid hormone in the blood is another cause of rising cholesterol levels.
The Framingham Heart Study is the longest running heart study and began in 1948 and is still ongoing today. In a 1992 editorial published in the Archives of Internal Medicine, Dr. William Castelli, a former director of the Framingham Heart study, stated:
"In Framingham, Mass., the more saturated fat one ate, the more cholesterol one ate, the more calories one ate, the lower the person's serum cholesterol. The opposite of what… Keys et al would predict…We found that the people who ate the most cholesterol, ate the most saturated fat, ate the most calories, weighed the least and were the most physically active."
The most important thyroid test is for Free T3. Free T3 largely regulates metabolism and many other body functions. Scientific studies have shown that Free T3 correlated best with hypo symptoms, while Free T4 and TSH did not correlate at all. In the future you should always make sure they test for both Free T3 and Free T4.
From your limited test results, I'd say that your Free T4 is much too low in the range. I expect that a Free T3 test would show similar results. Free T3 in the lower half of the range is frequently associated with having hypo symptoms, like the ones you mentioned, including high cholesterol.
Be aware that a good thyroid doctor will treat a hypo patient clinically by testing and adjusting Free T3 and Free T4 as necessary to relieve symptoms, without being constrained by resultant TSH levels. You can get some good insight into clinical treatment from this letter written by a good thyroid doctor for patients that he sometimes consults with after initial tests and evaluation. The letter is then sent to the participating doctor of the patient to help guide treatment. In the letter, please note the statement, "the ultimate criterion for dose adjustment must always be the clinical response of the patient."
http://hormonerestoration.com/files/ThyroidPMD.pdf
So I suggest that you should go back and get tested for Free T3, along with Free T4. I think you should also discuss with your doctor that you are having hypothyroid type symptoms and would like to start on a therapeutic trial of thyroid meds to see if symptoms are improved. You might also show the doctor a copy of the link above and ask if he is going to be willing to treat clinically, as described. If not, you will need to find a good thyroid doctor that will do so.
It is also reported that having adequate Vitamin D is important for good thyroid function at the cellular level. So, you need to supplement to raise your D level to around the middle of its range. Also hypo patients are frequently too low in the ranges for Vitamin B12 and ferritin, so I suggest testing those as well.