These are all great questions.

The answer is — it depends on what you're cooking. For normal sautéeing, if you don't want to use Teflon or similar non-stick pans, then the best option is probably well-seasoned carbon steel. These pans are relatively light, strong, and if you take care of them they are very non-stick and will last forever. They need to be seasoned by first cleaning them very well (many manufacturers coat them in wax so they don't rust). Cooking potato peelings and salt in them until the peels are black works well (stir continuously). Then, subject them to very high heat when they are dry and with no oil. This is called "blueing," and it oxidizes the metal, turning it slightly blueish (unless they already come with this done, as many do). 

Teflon (and similar) pans have a reputation of being bad for health, but if you cook with them at medium and low temperatures there is no danger. They shouldn't be scratched simply because that ruins the non-stick coating, but if you do scratch them and some of the Teflon gets in your food, it won't hurt you. The problem is cooking in them at very high heat, which vaporizes some of the Teflon, and if you breath it in, you can get what's called Polymer Fume Fever, which is a temporary ailment that feels a little like the flu. Normal use and care of them does not pose health risks.

Some foods do not need oil (for example, you can cook down mushrooms in water without using any oil), but others (such as eggs) are very difficult to cook (e.g. over easy) with no fat whatsoever. 

Stainless steel pans wear well, but they are not as non-stick as seasoned carbon steel. Cast iron can also be seasoned and be somewhat non-stick (using oil), but they're not as good as carbon steel either (but better than stainless steel). Ceramic coating is also somewhat non-stick, and better than stainless steel as well, but not as good as carbon steel. 

Butter is an excellent fat for non-stick cooking of things like eggs, and it is more non-stick than oils like canola, olive oil, etc. For high-heat cooking, clarified butter or ghee are great. But avocado oil or other high-heat oils work well and are better for you. 

You asked about cleaning, and it is true that you should let pans cool down before cleaning. The reason for this is because cooling a hot pan quickly (e.g. with water) can warp it. People say you shouldn't use soap with cast iron, but that's not true. You just don't want to use harsh cleaners, Brillo pads, etc. that will remove the seasoning.