Odd question. Anyone know if the emulsification properties of the egg yolk are present when it is cooked? Meaning, could I use the yolk out of a hard boiled egg to bind oil and water in a blender or to mend a broken sauce? Not that I can think of an immediate use of such an idea, but I'm curious.

Depends how coagulated the yolk is. If it's chaulky, probably not, except perhaps crushed and used to start a mayonnaise. If it's runny or soft, you can perhaps use it to emulsify cold or hot sauces. A true Caesar Dressing is emulsified with a coddled egg.