Your TSH is high so it does look like you have Hashimoto's which is an underactive thyroid. It's diagnosed mostly on TSH and T4 initially rather than antibody tests.
The important blood tests you want to have done are TSH, freeT3 and freeT4. Ask your doctor to add in freeT3 to your blood tests. T3 is the active version of T4. Antibodies are also useful to have. If they are high they will confirm the diagnosis (but there is a 20% chance you can get a false negative with this test). If they are not high it's still useful because it can be used to compare against future tests.
Having an untreated underactive thyroid can give you high blood sugar and high cholesterol so that's why those have been ordered.
Extreme tiredness and weight gain are classic symptoms and loss of periods is too (although people more often get heavier periods, it is a symptom and I have no periods too because of it).
I think it would be helpful if you looked up full lists of symptoms as you may have other symptoms. It affects the whole body so there is a long list. Just look up something like "checklist of hypothyroid symptoms".
You've taken a great step by coming to this board. It's given me a lot of useful advice.
Thyroid medication will take a while to work. It usually takes 3 weeks to notice a difference and it takes 6 weeks to get into your blood so they can test your levels again. It can take some time to get the right level for you and in reading this board you will find many people trying to get their doctor to get the right dose for them, the optimal dose or set point, rather that just getting them into the "normal range". The good news is that you don't really get side effects with thyroxine. It's the same drug as your body would make so you only really get side effects if you take too much.
It will take a while but you will get better