I have a Big Green Egg myself. They don't use standard charcoal briquettes, but lump hardwood charcoal. See the link below for more info.
The recommended way to extinguish one of these grills is to close off all of the vents and let it starve itself of oxygen. The exterior of my grill will stay warm for about 24 hours after I finish cooking. I've never opened it up to test. I know of anecdotal cases where somebody disposed of the ashes from their fireplace up to two weeks later and had it cause a fire in their garbage can. Can't say I've heard of anything specific to a grill, though.