1. I'm not sure about this question. She should ask her doctor about it.
2. The side effects mostly come when the dosage is off, like when she has too much or too little. It sometimes takes time before they get the dosage right for the person and there is adjustment time of 2-3 months in between dose changes. So it can take time. Also, other things, like gaining or losing weight can change what dosage she will need. Some people do say that the supplemental pills don't works as well for them as their regular thyroid did, so they might be candidates for T3/T4 combination therapy, but I recommend that people try the T4 alone first, since many people are fine with that. If after she is on a "correct dosage" according to the blood tests and she doesn't feel right, then it may be time to explore other options. Some people also say that they gain weight, regardless of what dose they are on, but regular physical activity can help this a great deal.
3. It's not the thyroid that distributes the calcium intake (did the doctor really say that?!). The *parathyorid* glands do that and those shouldn't be removed if the surgeon knows what he is doing (unless those are cancerous). One or two may be removed because they may be imbedded in the thyroid gland or just because of how small they are, but there are four of them and it is very rare for all of them to be removed unless all of them are imbedded or the surgeon really doesn't know what he is doing.
She should take calcium supplements, however, because usually they want patients after thyroid cancer to be on a higher dose of thyroid hormone that is slightly hyperthyroid and this can cause calcium depletion and lead to osteoporosis.
If she gets a diet high in calcium, maybe she would be okay, but supplementing that would still be good if the doctor recommends it. However, if her parathyroid glands were removed during surgery, she will need supplemental, quick acting calcium several times a day to keep her blood calcium levels up, otherwise they could dip dangerously low. We don't just need calcium for our bones, but also for nerve function. It's vital.
4. I don't usually give a total recommendation, but given that your mom has such a strong family history of thyroid cancer and the fact that she is almost 50, I would think that removing just half wouldn't be a very good idea. Age generally isn't a positive factor in these things. It also depends on how large the nodule is (hers is small), but if she doesn't want to have to worry about it anymore. She also wouldn't be able to have RAI treatment unless she had a total thyroidectomy. They would be able to give her yearly or bi-yearly ultrasounds, but not the blood tests for thyroglobulins that track thyroid cell growth (that wouldn't be useful if there is still thyroid tissue).
I hope this helps. I wish your mom the best. Please re-post if you have more questions.
(Also, how can a healthcare system be superb if the doctors are mean and unhelpful? That's most of the quality of a healthcare system, if you ask me, how much the doctors care about and listen to their patients.)