No, you won't become hyper if you quit gluten. While some naturopaths and functional medicine doctors say that gluten causes Hashi's, there's no real, conclusive scientific evidence that this is true.
Your gut microbiome, is very important, though, so if you're only eating protein and vegetables, you're likely to be come nutrient deficient, very quickly since there are some nutrients that the body can't store that you might not be getting.
No, going gluten free will not cause you to become super hyperthyroid from the Synthroid.. There is a theory that the combination of "leaky gut" and gluten intolerance results in antibodies that also attack the thyroid gland, eventually causing hypothyroidism. This theory is only supported so far by anecdotal information, not scientific evidence. At any rate, whatever damage has been done to the thyroid gland will not be repaired, so that requires thyroid med to provide enough total thyroid hormone. That is why the doctor has you taking Synthroid.
You should be aware that the thyroid hormone that is metabolized by all the cells of your body is called Free T3. Most Free T3 comes from conversion of Free T4 into Free T3 when it enters cells. Conversion is affected by a number of things, so you should make sure the doctor always tests you for both Free T4 and Free T3 each time you go in for tests. Insist on both. Ideally your doctor will prescribe enough thyroid med to relieve hypothyroid symptoms. Symptom relief should be all important, not just test results. If your body is not adequately converting the T4 to T3 then you may need a source of T3 added to your med.
Also, hypothyroid patients are frequently deficient in Vitamin D, B12 and ferritin. So if not tested for those you should do so and then supplement as needed to optimize. D should be at least 50, B12 in the upper end of its range, and ferritin should be at least 70.