Is it okay to store cooked omelette for 1 week?
Secondly, at what temperature it should be stored ?
You could keep it frozen for longer than a week, but by the time you reheated it, it would be like rubber.
I wouldn't attempt more than a couple or three days in the fridge.
The only way I can think to keep anything omelette-like would be like a Spanish tortilla - maybe it's the potato & onion that changes the texture, but that is fine after being cooled then re-heated.