Yes it is extra to feed the dolphins, I think I remember it being around $7. There are a couple other animals you can feed as well, the rays and something else... Sorry drawing a blank at the moment.
You can feed Stingrays, Seals/Sealions, and small Sharks. The dolphin feeding is $7 per tray and each person entering the feeding area has to buy at least one tray.
If kids are three and under you don't have to buy a tray. My son was barely four and they didn't make us buy him one. My daughter didn't want to touch the fish so my son fed them more than she did. If you are an AP holder I think you get a discount too.