You can, and people often do this to cut cholesterol and fat from their diets. Egg whites alone still possess the coagulant properties associated with eggs, but they do lack the flavor, color, and richness found in the yolk. I would recommend using part whole eggs, part egg whites to find a compromise between the two.
To solidify and set the dough. Egg yolks enrich a dish, egg whites set/solidify a dish. If you look up a recipe for Consomme on Youtube you'll see a blatant use of egg whites and their natural setting agents.
Yes, you do use egg whites
If a recipe calls for "2 eggs," it means THE WHOLE EGG. If they want you to use the egg yolk only, it will say, "add the yolk of 2 eggs." If they want you to use the egg whites only, it will say, "separate the eggs and beat in the whites." To 'separate' an egg means to separate the yolk from the whites.
You would use 4 medium egg whites or 5 small egg whites.
if you want to
It is a richer custard if you use just yolks. But you can use more yolks than whites and it will still be yummy.
Only if you want salmonella and anal leakage AKA diarrhea
If you are looking at decreasing cooking time you would use cold water. Thermal exchange causes the water temperature to have a drastic increase. If you are just cooking the egg then it doesn't really matter.
Make merangues.
Any bowl or utensil that is used when beating egg whites has to be perfectly clean and grease-free. When air is beaten into egg whites, the proteins of the egg white open up, which makes the whites form a semi-solid. If there are any lipids introduced into the egg whites they prevent the whites from forming a semi-solid state by surrounding the proteins of the egg whites and preventing the proteins from opening.
The only substitution for egg whites in a frosting made with egg whites would be powdered egg whites sold in baking supply shops. Powdered egg whites have the advantage of being safe from salmonella sometimes found in raw eggs.
Omit the yolks, and try using two egg whites per egg called for. The extra egg white will make up for the moisture lost in the omitted yolk. For instance, if the recipe calls for two eggs, use four egg whites.