How to get rid of mice: Personally, I go with squishing. After the first couple, I got a whole lot less squeamish. You can get them off glue traps with cooking oil, but it's really not worth it. The Hav-a-Heart traps are probably best. A cat might work too, but would definitely not be non-violent. Squishing's probably more humane.

Regarding traps disappearing - sometimes they don't land on a part of the mouse necessary for the continued existance of the mouse, and the mouse drags the trap elsewhere. Your friend will eventually find it, perhaps with mouse attached.

As to the nasty new smell and "Could a mouse have died somewhere?" Yes.
"how to find & remove the corpse or will it, eventually, stop smelling"
They always stop smelling eventually, and a new one takes its place. My own opinion is that it's impossible to find the little corpses when one is looking for them, and they have a tendency to die in places humans don't even consider looking and can't get to anyway. Mice are small, though. It usually doesn't take too long before they become little mouse-mummies.