@kornboi I do have issues with the recipe, as I noted on the second page of it. However, adding various condiments to beaten eggs before cooking is a fairly common practice. The most frequent things you'll see recommended are plain water, heavy cream, milk, or mayonnaise, as they used here. Eggs mixed with such condiments are purported to be lighter, fluffier, or creamier, or possibly even hold all three of these properties at once. I myself use cream, and occasionally a little mayo. One person who adds mayo to his eggs is notable American food presenter Alton Brown.
Regarding the pan: since you're supposed to have emptied it out and then cooked the rice in the other pan with all the other ingredients, it should have cooled down a little. As a side note, I would recommend a non-stick pan if you're not great with eggs, since you need a good flip to get the omelette onto the rice. An extra bonus is that it's harder to char them in a non-stick pan.
PS Never use metal utensils (or those made out of other hard materials) on non-stick pans, as they could scratch the coating. Scratched non-stick pans, in general, leach a lot of terrible chemicals into the food.
Edit:
@rei893 Did you seriously?