Purpose of cooking oil
I'm going to only answer the bit about oils in cooking, rather than all the uses of oil in the kitchen.

When frying or sautéing, oil acts in several capacities:


First, and most critically, the oil in a pan conducts heat from the hot pan to the food being cooked. The oil greatly increases the surface are of the food in contact with heat, rather than have just the bits directly in contact with the pan get hot.
The oil prevents (sort of) food from sticking to the pan or pot.
The oil (indeed any type of fat), is an important flavour carrier. It makes food taste better by bringing out the flavours.
Oil facilitates the Maillard reaction, which is how we get the lovely fried crust on the outer surface of fried foods.