Sounds like a day of bonding.
I have never done eggs, let us know how the turn out.
Smoked Paprika is really good.
As far a cold smoking in the grill, if the top does not have an exhaust vent, I would vent it by leaving the hood open about 3/4 of an inch.
If you find a large enough cardboard box you could even place this over the open grill with a couple 1" in the top. The idea is to allow the smoke to pass around the food to flavor it as it passes through the smoking chamber.
Good Luck and keep us posted.
Humphrey's DownEast Beast W/BBQ Guru