This depends on which type of omelet you want to make, I'll run through the three types I know how to make.
The thick Waffle House style omelet (it's the kind that poofs up and is about an inch thick all the way around, IHOP also serves this omelet) is achieved by beating the eggs and incorporating a LOT of air in to the mixture. Restaurants do this by using a milkshake machine to whip the eggs. This type of omelet is easy to flip and cook, you just put enough butter to coat the pan, pour off any excess and then pour your egg mixture in to the pan (you are going to have to use an omelet pan for this). Let the omelet cook while occasionally swirling the pan in a circular motion. When the omelet has cooked most of the way through flip by either the toss in the air and catch in the pan method (not for beginners!) or the fork and spatula method (place spatula underneath and hold omelet onto the spatula with a fork on one corner). This will yield a thick fluffy omelet but it can be hard to work enough air into the eggs at home.
The second type is the French omelet and it doesn't require flipping at all. The french omelet is what you see in fine dining brunches if you see it at all in the U.S. This omelette is made by mixing your eggs and pouring the mixture into a medium low heat pan. As it cooks you take a fork and pull the cooked egg off the bottom and into the center of the omelet. Here is a video that goes more in depth.
The third type is what I call the grandma omelet. It's made in a frying pan instead of an omelet pan and features a fairly flat egg wrapped around some cheese and other toppings. For this omelet you need medium heat. You should coat the pan with butter and pour in the egg mixture to a heated pan. Let cook for a minute or two, then swirl the uncooked egg to the outside of the pan use a fork to lift the very edge to allow the uncooked egg to take the place of the cooked next to the pan. Let the egg cook till desired doneness (side note: your omelet should and will have a small amount of uncooked but heated egg, it should be not runny, but still not be entirely set) and add toppings and fold over. Then serve.
To address your scrambled egg problem, you need to let the egg set more before attempting the flip and lowering the heat on your pan will take care of the burning. Just remember that eggs are extremely delicate and respond better to lower heat and close attention.
Also the names are just the ones I use in my head, if somebody knows the correct terminology, let me know and I'll update to reflect it.
There isn't a standardised shape as such, but the rugby ball shape is common. To achieve this shape, all you have to do is cook your omelette (French style uses a super-hot pan and lots of butter), then roll three quarters of it up in the pan.
Then nudge the omelette up the side of the pan a little, so that it partly protrudes over the edge, which will then allow you to flip the remaining quarter back over onto the rest of the omelette. You should now have a rugby ball shape with a 'seam', i.e. the edge of the omelette - simply turn this out so the 'seam' is on the plate and you have a nice smooth omelette showing on top.
Best Answer
I have a theory about your omelette (and sadly, no means to test it). But I think it is plausible, please feel free to point out logical errors.
I think that the unappetizing coloring is due to the creation of sulfur salts in your pan. Egg whites are rich in sulfur, as are all alliums (including chives). If you heat them enough (you don't mention whether you overcooked the omelette, but it happens often enough), the heat changes the molecules in which the sulfur is bound, and the sulfur atoms (or sulfur containing ions) are free to react with whatever they find around. If they happen to find iron or copper, they form sulfides and sulfates with them. These sulfides and sulfates have colors ranging from blue to green, and pure iron sulfide is black. A mixture of them with the yellow yolk can easily look gray, especially when the color is desaturated through the addition of white cream cheese.
You say you used a coated pan, but there are still possible sources for the iron and copper. First, nuts contain trace elements, including iron and copper. Second, yolks too contain iron (in fact, the bluish coloring on the surface of a hard boiled yolk is an iron sulfide created when the proteins in the egg white denaturated from the heat). Third, you can't exclude contamination of the ingredients (plants sometimes store metal ions from the environment).
I have no way to prove any of the above, but at least it sounds like a good working theory. But I will understand if you aren't eager to reproduce in order to investigate the effect closer.