Is it possible to have another FNA? Before my FNA, I was told that if it came out indeterminate, they could repeat it or I could have the surgery.

I don't know if you can read anything into the family history. As far as I know, only one form of thyroid cancer is likely to be genetic and that is one of the less common forms.

You've probably already read this, but I will say thyroid cancer is generally very slow acting. I probably had it for 5 years or more by the time it was diagnosed (certainly, now that I know what I am looking for, I can see the nodule in photos from 3 or more years before my diagnosis) and when I was diagnosed, the consultant was like "well, it's not going to do anything in the immediate future, but psychologically, it's probably better to get it dealt with as soon as possible" and the nurse said it was better to remove it now while I was young and healthy. So they weren't exactly in a hurry. And my nodule was 10.5cm.

My operation was a lot bigger than yours would be, if you choose to have it, as I had my whole thyroid and a number of lymph nodes removed, but I didn't have much problem with it. Different people do have different experiences though.