These are all great questions. I'll start with the oil questions. When sautéeing, oil does a number of things. It keeps food from sticking to the pan, it provides heat contact with the pan (cooking more evenly), it breaks down into other molecules (this is usually a bad thing), and it carries fat-soluble volatiles, which may be in the oil already or may be dissolved into the oil before and/or during cooking (for example, by infusing spices before adding the chicken — a technique called tadka). 

If you want to try an experiment, take a Brussels sprout and cut it in half, then sauté one half for 10 minutes over medium heat in a tiny bit of oil (spray the pan with cooking oil) and then sauté the other half in 1/4" of oil, again for 10 minutes. They will come out very different. The first will be pale and unevenly cooked, and the second will be evenly cooked and much darker. 

When cooking chicken, you can use less oil and still get good heat contact by pressing the chicken down against the pan. A very old technique is using a brick (but anything heavy will work). Take a brick, cover it in clean foil, and put it on the chicken. The chicken will make more even contact with the pan and will cook better with less fat in the pan. If you are sautéing a chicken breast (which has less fat than dark meat), you should cook it with the skin side down (assuming you're leaving the skin on, which you always should as it helps insulate the chicken from overcooking and drying out), and you may want to poke holes in the skin or make small cuts so the chicken fat renders out as it cooks. That will make the skin more crispy, and you'll also end up eating less fat. If you don't want to eat the skin, take it off after cooking.

A non-stick pan will allow you to cook the chicken more easily with less fat. The only caution is that you can't heat up non-stick pans nearly as much as normal pans because the chemicals that make the nonstick surface can vaporize and cause temporary illness (polymer fume fever). This will also hurt the pan. 

The "air fryer" (just a small, high-velocity convection oven actually) is a great tool for cooking with low fat and getting good results. It won't taste as good as cooking in fat, but few things do. Except ice cream. 

Figuring out exactly how much oil you are eating will be tricky because the pan will be left with fat and other compounds from the chicken after cooking, so there will be no way to accurately measure the amount of oil left in the pan. 

Also, the reason your (white meat) chicken gets dry is because it's over-cooked. We'll talk about this a lot in class and a number of things you can do to keep it moist and delicious. 

However — if you're worried about fat consumption, there is a much better way to cook the chicken, which I will actually be showing in class later in the quarter when we have our proteins lecture. That method is poaching. I will show you a couple of great techniques and also some ways to perfectly season the chicken. 

Note: I just watched the video, and funny that he used the same example of Brussels sprouts. There are a number of scientific mistakes in the video, however. For example, oil is actually a terrible conductor of heat, relatively speaking (though much better than air). Also, Maillard browning reactions happen even at room temperature, albeit very slowly, and caramelization temperatures depend on the sugar. 

We will discuss all of these things in much more detail in class.

Chef Michael