> 1. Does a hard boiled egg contain oxidized cholesterol?

No (negligible).

> 2. Does an omelet contain oxidized cholesterol?

Yes (lots).

"Avoid Eating Lipid Peroxides and Cholesterol Peroxides

Remember, one of glutathiones' primary jobs is to destroy peroxides before
they can cause any damage, and whenever a peroxide is destroyed, a
glutathione molecule is neutralized.  We produce plenty of our own peroxides
that require destruction without bringing any on board via our diet.  Those
that we do consume deplete our glutathione.  Where do we get lipid peroxides
in our diet?  All kinds of places, some obvious, some not so obvious.

One common source is scrambled eggs.  Egg yolks contain a large amount of
cholesterol that is easily oxidized when subjected to heat and air.  When we
scramble eggs we break the yolks and expose the cholesterol within to both
heat and air., producing a slurry of cholesterol peroxides.  Although for
Hedonists and Dilettantes the occasional omelet will do little harm, we
recommend that most of the time you adapt a Purist's approach and eat your
eggs poached, boiled, or fried in such a way that the yolks remain intact,
protecting their cholesterol contents from peroxidation."

source: The Protein Power LifePlan p125 - 126

Rob