Thyroid removal (6 growths, 1 possible cancer) and taking synthroid (thyroxine). Can this help regulate for weight loss and facial hair issues? I can't lose weight.

NO , Purpose of taking thyroid hormone (synthroid (thyroxine) ) is a replacement exact amount needed , due to lack of production as gland was removed by surgery , which controls metabolism of every cell in the body , other issues or associated hormones may be causing your problem , speak to your doctor will help you .
Yes. Hypo(=low)thyroidism can lead to weight gain. Normalizing the thyroid level can regulate those hormones and lead to normal thyroid level weight. Obviously weight gain and loss is a balance of calories in and out from exercise and diet as well. Good luck.
Eat less. Your doctor should be able to advice you by checking your blood levels. If you need further help, go see a specialist;;;called endocrinologist, who can sort it out for you.