"Lady o' the house" writes:
I've been wondering lately if I should up my calories. Lately, I've been
averaging around 1200 calories and 30-35 carbs per day. I'm losing about a
pound a week (I know, that's great and I should be happy---52 pounds in a
year).

No reason to change anything if you're reliably losing weight.

Also, could the fact that my calories are low be a cause of my thinning
hair? It seems to have happened shortly after I began to really count
calories. I've always had really thick and full hair; now I can see my
scalp through the top front of my hair.

Stress can do that; dieting does cause stress. Of course, lots of
other things could be the cause too, you never know.

Or should I just do my own experiment?

Yes ;-)