I agree with Barb.
Some will argue that gluten free will help reduce the antibody count with regard to Hashi's. But as stated there is no real scientific evidence that is or is not true. And even if true, all it will do is slow the onset of Hypo due to Hashi's at the best. Meaning that probably the only benefit even if it is true will only mean that you postpone the need for Thyroid replacement medication.
Gluten may make people feel bad. And going gluten free for that reason is reason enough to do it. But I would not put too many eggs into the basket that it does much of anything what so ever for Hashi's.
You need to get blood tested for Free T4 and Free T3 at the very least and report yuour results along with the reference ranges for each as they vary from lab to lab. That will best help determine what your thyroid status is.
Thyroid is one of the primary hormones that controls your body's overall metabolism rate. If you have a low thyroid problem, your metabolism will be slowed and your body simply makes it difficult to impossible to burn calories. And thus weight gain. Which is why weight gain and inability to lose weight is such a common symptom of low thyroid!
You have a lot of questions here... I'll do my best to answer them... There's a lot of controversy over the gluten issue. First off, there is not real science that says that being gluten free will do anything to help anyone recover from Hashimoto's, though there are those that insist it will. As for weight, again, it's not gluten that causes weight issues, it's the number of calories one eats, as well as metabolism.
If your thyroid hormone levels are not adequate, your metabolism won't be adequate either, so it's likely that you will gain weight, whether you eat gluten or not. Insulin levels and the way your body tolerates glucose and carbohydrates also help determine your weight.
The easiest way for us to help you is for you to tell us what your current thyroid hormone levels are, so we can better assess the testing/treatment you are receiving. Be sure to include reference ranges with any test results you post, since ranges vary lab to lab and have to come from your own reports.