The basic differences between white and brown rice is explained below (as written here):
An important first question to ask about all rice-and for that matter, most foods-is how much it has been processed. In the case of rice, processing usually involves milling and polishing. The outermost layer of rice, called the hull, is removed to make brown rice. Brown rice is rice with the whole kernel intact and the kernel is still surrounded by all layers of bran.
To produce white rice, the bran layers of the rice have to be milled off. Most of the rice germ is also removed during this abrasive grinding process. At this point in the process the rice is called milled, unpolished white rice. Finally, a wire brush machine is used to remove the aleurone layer that remains on the rice. This step is called polishing. As polishing is not an all-or-nothing process, semi-polished rice may still contain parts of its aleurone layer.
Additional information about brown rice (here):
[Brown rice] has a mild nutty flavor, and is chewier and more nutritious than white rice, but goes rancid more quickly because the germ—which is removed to make white rice—contains fats that can spoil. Any rice, including long-grain, short-grain, or sticky rice, may be eaten as brown rice.
So, because they are processed differently and have different parts of the grain intact or removed, your method of cooking is going to vary.
I have used this method of cooking brown basmati rice with pretty good success. It includes washing the rice a number of times prior to cooking and adding salt. This method advises adding oil after the cooking process, but I typically add it into my rice while it cooks, or even sauteeing the grains in oil then adding water to the pot to cook.
It is also suggested that leaving your brown rice to soak for 15-20 minutes (and sometimes even longer) can be beneficial as well.
All of that aside: if you're looking to stick with a flavor and texture similar to that of white rice then brown rice may not be your thing. I know that as a jasmine rice eater I do not particularly care for the texture of brown (even brown jasmine). As mentioned above it does have a "nutty" flavor and is "chewier" regardless of how well/properly it is prepared. It changes the flavor and feel of every dish I serve it with. Just a heads up.
Once you get the hang of it, rice is as easy as pasta. One thing you say in the question that may be central to the difficulty you are having is that your lid is "half-closed". For the majority of rice cooking methods, not only should you keep the lid tightly closed, but you shouldn't even open it to check the rice until it has cooked close to long enough that it might be done. Look at the accepted answer here: Rice gets burnt and watery. That answer explains it more thoroughly.
Best Answer
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.