Q: What if my child accidentally makes a purchase, can we be refunded if we never watched a rental?

All purchases are considered final, but you can try the 'report a problem' link to contact iTunes Support and see if they will refund or credit you : http://reportaproblem.apple.com (iTunes Support should reply within, I think, about 24 hours).

If the 'report a problem' link doesn't work then you can try contacting iTunes support via this page : http://www.apple.com/support/itunes/contact/- click on Contact iTunes Store Support on the right-hand side of the page, then Purchases, Billing & Redemption