Omelettes are a cook-and-eat dish; they turn rubbery reheated. If you want eggs that store, make a frittata instead — same ingredients, baked, good for 4 days.
The pan wasn't hot enough when the egg went in, there wasn't enough fat, or the non-stick surface is scratched. Heat the pan, then add butter, then the egg.
Three for one serving in an 8-inch pan. More than that is too thick to cook evenly — make two omelettes rather than one big one.
American folds a filling into a set omelette, which is this recipe. French is stirred constantly for a soft, custardy interior with no browning. Same ingredients, different technique.