No, I didn't miss your point, at all... I understood, perfectly.
No, your body would not be reliant on the Synthroid, at this point; however, you can't live indefinitely, without adequate thyroid hormones, since every major system in your body depends on them.
If you have Hashimoto's your thyroid is constantly being destroyed by antibodies, so its function is declining every day. Today, you have menstrual and body temperature issues; next week, you might low heart rate, or brain fog, weight issues, stomach/digestive problems, acid reflux or body/joint pains, your hair might start falling out, you might not be able to get pregnant, or you might have a miscarriage or any number of hypothyroid symptoms.
It's much better to find a doctor that will test and treat you properly than to abandon the medication, completely.
I'm assuming that the level(s) you quoted (6.2 and 2.89) were for TSH... Is that the only thing that was tested? What about Free T4 and Free T3? Those are the actual thyroid hormones and are more important than TSH, which is a pituitary hormone and not indicative of actual thyroid status, so does not give the whole picture.
If you have results for Free T4 and Free T3 (both before and after starting medication), please post them, along with their corresponding reference ranges, so we can see your actual thyroid hormone levels.
The dosage you're on, 25 mcg, is very small and what it does is simply lower your TSH, so it won't stimulate your thyroid enough to produce as much natural hormones as it might have been previously, but it isn't enough to completely replace what your body actually needs.
The fact that your menstrual cycles were off and your TSH was higher than normal is an indication that there's a problem and more symptoms would be appearing soon. That said, you need to be dosed by actual thyroid hormone levels, not just by TSH.
It's not unusual to feel worse when first starting a thyroid hormone medication, however, dosages do, often need to be adjusted after a short period. The problem is that your doctor is going, only by TSH, not actual thyroid hormone levels. Doctors who dose their patients, based only on TSH tend to keep their patients ill for long periods of time.
Ask your doctor to order Free T4 and Free T3 and if he refuses, it's time to get another doctor. In fact, I might advise you to do that anyway, because doctors that have to be asked for Free T4/Free T3 usually tend not to be very good thyroid doctors anyway.
Have you been tested for thyroid antibodies to determine if you have Hashimoto's Thyroiditis? If you have Hashimoto's, you will require the medication for life and the sooner you get started, the better. The tests you need to determine if you have Hashimoto's are antibody tests, Thyroid Peroxidase Antibodies (TPOab) and Thyroglobulin Antibodies (TgAb). You need them both, as some of us have one or the other; some have them both.
When taking thyroid hormone medication, it doesn't really matter what time of day you take it, as long as you take it, consistently, at close to the same time every day.
It's not always easy to figure out a schedule for taking medications, but we find a way, once we realize what a difference it makes in our lives and how important it is. Since it hasn't made you feel better, you aren't seeing the need to make the accommodation yet.
I forgot to mention that my current dosage of the Synthroid is 25mcg.