Absolutely. Typical thyroid cancer in women less than 50 has about a 98% survival rate. Although there are occasional very aggressive thyroid cancers (mostly in older men), most thyroid cancer is easily treated. Make sure you find an experienced thyroid surgeon. Try the american association of endoctine surgeons website.
Yes. There are several types of thyroid cancer and the treatments are different. The most common is papillary cancer and it is treated with surgery and thyroid hormone replacement. If there is a high risk of spread then radioactive iodine is given after surgery to 'ablate' any remaining thyroid so that, if there is a recurrence it can be detected early and treated. Then recurrence is unusual.