Is it possible if the thyroid gland quits working completely before age 3 that other glands can be permanently affected as well?

Unlikely but... This is complicated! Even mild hypothyroidism can have permanent effects on a child's growth & development, but it's not likely to affect the function of other endocrine glands. However, hypothyroidism MAY be accompanied by dysfunction of other glands (multiple endocrinopathy) or it may be caused by dysfunction of the pituitary gland. I assume your doctors have checked for these other problems.