Does a fibroid ever go away on its own?

Yes. Occasionally, usually after a full term delivery, a fibroid may undergo degeneration due to lack of blood supply caused by the uterus' major shrinkage in size immediately post partum. If the fibroid gets no blood, it stops growing and dies. Natural mechanisms then cause the dead tissue to be re-absorbed and the fibroid "goes away on its own". In general however, they don't go away.
After menopause. Fibroids can fluctuate size during different hormonal times (ie: pregnancy). In general, fibroids can be present and potentially grow during the reproductive years. After menopause, when estrogen levels have decreased, fibroids will gradually shrink and can eventually become undetectable. Fibroids should not grow during menopause. Hope this helps. .
Yes. Fibroids are a result of high estrogen levels. High estrogen can be caused by excessive carbohydrate intake (via competetive inhibition of glucoronidase enzyme) therefore a low carb diet is beneficial to reduce estrogen and help shrink fibroids. You may also have low iodine, thyroid problems, low progesterone. All of these things should be evaluated. All phytoestrogens should be avoided.