Do prenatal pills really make skin clearer and hair grow?

They can. Your body requires certain vitamins and minerals for proper metabolism and for maintenance of healthy tissues. If the vitamins are filling a deficit in your diet, they will potentially have the effects you are questioning.
Somewhat. Prenatal vitamins are those vitamins that help cellular replication, and with a growing fetus, these vitamins can sometimes get used up. There is also rapid cellular replication in the skin and hair, and so taking these vitamins will help skin and hair. However, if you already have adequate levels of these vitamins, you probably won't see much difference.
No, hormones do! The hormones that are responsible for maintaining pregnancy (estrogen and progesterone) are what cause changes in your skin and hair. When you deliver, your skin will most likely return to normal, and you may experience some hair loss.