Absolutely. With appropriate treatment there is no reason full recovery would not happen within a short period of time.
Yes. Yes, it's possible that an individual be diagnosed with hashimoto's thyroiditis only to be serology negative a few months later. This is however, the exception rather than the rule. Talk with your doctor if what you had was acute thyroiditis due to infection instead.
Most PCPs and endo. Most internists and endocrinologists are familiar with the disorder. It is an autoimmune disorder that often leads to an under active thyroid. It is the most common cause of low thyroid hormone levels. There is no treatment for hashimotos per se, but the thyroid hormone levels can be kept normal with medication.