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.
The coating you are talking about is potato starch that is browning on the bottom of the pan, similar to what happens to roux when it is prepared. If you deglaze the pan using alcohol, it will come right off without any effort (water works too, though more is needed).
As for how to get the potatoes not to stick, it's important that the pan and the oil are quite hot when the potatoes are added to the pan. As described at Newton Ask a Scientist, the hot oil encourages a steam barrier to appear between the potatoes and the pan, temporarily protecting the potatoes from sticking and encouraging the formation of a crust that will permanently keep them from sticking. Depending on the amount of potato being cooked, the amount of oil added to the pan will have to be adjusted for thermal mass, since the addition of uncooked potatoes will cause the oil to cool down very quickly.
Best Answer
Oil or fat is absolutely not necessary to cook rice.
I suspect you may have been taught the pilaf method where the rice is first sauteed in oil or butter, and then liquid is added and the rice is fully cooked. The purpose of the pilaf method is to add depth of flavor. When making pilafs, additional herbs, spices, or aromatics (such as onions) are often included.
If you have been taught to add oil without the sauteeing, then the only purpose of it would be to add flavor. It won't really help prevent sticking, and isn't needed for the rice to cook properly.
There are many methods to cook rice in addition to the pilaf method, and many to do not require any fat. Simply boiling or steaming in water is completely traditional in many Asian cuisines.
In some cuisines, the crusty rice that sticks to the bottom of the pot or rice cooker is actually highly prized (see Wikipedia's nurunji article).
You can mitigate the crusting of the rice by making sure to use the lowest flame or heat setting that will keep the rice steaming. I actually prefer to put the whole pot in a moderate oven, rather than do it on the stove top.
You can also use the pasta method of simply boiling the rice in water, and then draining it when it is done to your liking, to completely avoid the crust. This will result in less sticky rice since the surface starches will be left in the boiling water. I understand this method is popular in some parts of India and Pakistan.