Why does the skillet burn even when using peanut oil

Part one:

Yes, put oil in first then put on a low/med heat. Generally cast iron should be heated gently, depending on your stove it should probably take about 5 minutes to get to a good temperature. The oil shouldnt evaporate - I've never experienced this. As to the type of oil - I use various depending on circumstance vegetable or olive usually.

Part two:

I generally shake the pan gently - when the oil becomes more less viscous (more runny) then it is at a good temperature to start cooking. You can also put something like a small piece of onion or bread into the oil - if it starts bubbling and generally cooking then its good to go. Some people use a couple of drops of water to see when it spits but I dont like doing this partly due to the safety factor and also due to the fact itll splatter over the stove top.

It depends a little on what you are cooking - if you are looking to brown some meat for example you'd probably want it a little hotter than if you were just going to make vegetable soup.

I feel the same as you Anderson. I found that too and you just made me to do some research, which I found they have different smoke points.

Unrefined sunflower oil 225°F 107°C Unrefined peanut oil 320°F 160°C

see the link: http://www.cookingforengineers.com/article/50/Smoke-Points-of-Various-Fats

Also, one of the reasons food stick to the pan is the evenness of the temperature of the pan. If you use a thicker pan or copper pan, the heat will be distributed on the pan more evenly and food will stick less. The higher smoke point of oil makes me to believe the oil helps to distribute heat more evenly.