I did this I have it too.
I took one years worth of medication with me so I wouldn’t have to think of it the first year I was here. After we got married I was added to my husband’s insurance and the medication have not been expensive at all.
No the children don’t naturalize only you do. They get us citizenship because you as their parent become one.They can not change name with your citizenship certificate and cannot do it with us passport or N-600 without court order.