What would be the best order for folding whipped egg whites and yolks into cookie dough?

It all depends on what you want the whipped egg whites to accomplish. Generally you use the wipped-egg-white-method in order to incorporate more air into the batter, which is mostly used for cakes. In this case you definitely want to add them at the end, so the air stays in the batter.

For cookies, this is probably not necessary, because you can't incorporate much air, as the batter is much thicker than a cake batter. However, depending on the exact recipe, you could want to beat butter and sugar until airy and fluffy. I generally first cream the room temp butter, then add the sugar, beat both fluffy, add the whole egg and beat further, then add the flour, baking soda, and other ingredients, and fold in. If your cookies are made from a even thicker dough, on the other hand, you can just dump all your ingredients in and just mix or knead them together to form the dough. In this case you'd probably want the butter cold, though. In both cases: don't overdo the dough. And don't overthink it.

The leavening generally comes from the baking soda. The freshness of the egg can also have an effect on the leavening, I suppose, but I'm not sure how much of a difference it makes. This effect is more prominent in cakes that contain a lot more egg than cookies do.

(Side note: For cakes, my general method is to first beat the butter and sugar until fluffy, add the egg yolks and beat further. Then add (sifted) flour and other ingredients. Finally, I add the whipped egg whites in at least two, mostly three batches. So first add about a third of the egg whites, incorporate them (you don't have to be too careful here, it is only to lighten the batter), then add the second half and fold in, then add the third half and carefully fold in. Bake immediately. )

This methods give me the best results.

Additional notes:

• if you want to use whipped egg whites, always remember, that egg whites don't like fat. This is the reason why you separate the eggs before beating, because egg yolks contain quite a nice amount of fat, which prevents the whites from whipping up.
• You can stabilize egg whites using sugar. However, be careful not to add to much of it add once, or the egg whites will collapse. I also noticed that the whites collapsed more easily when using powdered sugar. My explanation is, that the sugar crystals are much smaller in powdered sugar and they can destroy the egg white bubbles.
• Whipping egg whites with a pinch of salt and at room temp will make the whipping a lot easier and faster.

Additional, additional notes:

• If airy cookies are your thing, you should definitely check out macrons. Although the french ones are the most popular, there are several other types, that have a meringue as a base, gently mixed with ground nuts or other things. They generally don't contain much fat, except for the fat contained in the nuts or whatever. For a personal favorite of mine, search for "german hazelnut macarons" on google. :)

Do you have a recipe? I'd follow that..

That said, this sounds like a lot of cakes I've made. Usually you cream the butter and sugar, then add the eggs to that. Finally you add the flour mixed with leavening, not mixing too much, and toss it quickly into the oven. That gives the best use of leavening.