Usually not. Some pans can take higher heat than others, but "stainless' and "non-stick" is not enough information to be specific.
No. At least, almost always not. The major exception comes when rendering fat, as in cooking bacon. There are others.
Yes. At least, almost always yes.
Yes.
No. The heat should almost never be on full for a home cook, unless heating a very large pot full of liquid. It should not be full up for preheating the pan, and/or the oil. You should estimate the setting you'll use to do the cooking when the food first goes on the heat, and go with that. For almost all sauteing, searing and frying, the choice is between medium-high and medium. It may take you some practice to recognize medium-high on your particular stove.
It depends how you're cooking them, i.e., fried, omelette, scrambled. Fried eggs and omelettes are usually cooked at around medium heat, while scrambled eggs are cooked at a slightly lower heat. In either case, it's important to let the butter foam subside before adding the eggs to the pan.
Watch your temperatures to keep from burning the butter. Just like overcooked toast, it's not good.
Hope this helps,
BDL