You can't fight while riding a chocobo, in fact it's possible for Chocobos to just abandon you and run away when monsters are spotted. There are different types of veggies in the game you can feed them which will increase their intelligence and morale to help avoid this. Black Chocobos are still being worked on but are apparently very special as you have to catch them in the wild.
I was wondering about this. I'm glad to hear that they're in the game.