Don’t know anything about the thyroid, but I can say that HbA1C and cholesterol are linked through insulin levels. All the diseases comprised in metabolic syndrome—cardiovascular disease, diabetes, gout, inflammation, Alzheimer’s disease, high blood pressure, PCOS, irritable bowel, etc.—are the result of insulin resistance and the consequent high levels of insulin in the bloodstream, caused by ingesting too much carbohydrate. HbA1C is a measure of how glycated your red corpuscles are, and because they only live for about three months, your HbA1C score is something of a stand-in for a rolling three-month average of your blood glucose levels.
Most people who go on a ketogenic diet experience an improvement in their cholesterol, both increased HDL and lowered LDL. Some do not, however, and Dave Feldman has worked out the mechanism by which these “lean-mass hyper-responders” can affect their numbers. Our innate cholesterol level clearly has a genetic component, as witnessed by people who have familial hypercholesterolemia: even though a group may all be related, they are not all eating the same diet, so there is clearly a heritable component to the cholesterol mechanism.
There is evidence to suggest that eating saturated fat improves cholesterol numbers, so a ketogenic diet can certainly help by increasing our saturated fat; the other part of a ketogenic diet, the low carbohydrate, will keep blood sugar consistently lower, which improves HbA1C.