Does hashimoto-caused hypothyroidism ever go away completely? Have you ever seen a patient like this who could stop taking thyroid pills permenantly?

Rarely. It can occur, but it's rare. Luckily brand name T4 replacement is cheap and safe.
No. If you have confirmed hashimoto's and already is hypothyroid, you will need thyroid replacement for life. There are some patients who have hashimoto's antibodies but have yet to become hypothyroid. Some of those might not become hypothyroid but will need blood testing monitoring for life.