How to cook rice & chicken simultaneously?
There are many dishes where chicken and rice are cooked mixed together in the same pot. Look at Arroz con Pollo for example, it's an easy classic and there are loads of recipes. There's also some types of Biryani, Jambalaya, and other traditional recipes from across the world that would work. 

Chicken pieces and rice cook in about the same time as you say, if you want to get vegetables in there it's a bit more complicated as you want the vegetables to be cooked, but not overcooked. I've generally made them separately to put put as a side as it's more attractive, however large carrot chunks cook about the same time, and you can put in pureed vegetables or even vegetable juice. One my mother makes uses crushed tomatoes instead of water to cook the rice in. 

As for having the chicken mixed in or on top there's no right answer, I like to brown the chicken and lay it skin side up on top of the rice as the rice will cook up around the chicken and the visual effect is good when the pot is put on the table. Visuals aside there's little difference in the method.