[Suhiir]

Mines represent most any sort of moderate size explosive, and can be detected/cleared more easily then Booby Traps or IEDs, and won't necessarily blow when a unit enters/exits the hex.

IED's compare to a 2000# bomb, truck size weapons buried as gigantic command detonated mines, that generally blow when any unit enters the hex, engineers don't get free ride!
I haven't tested yet but the new patch is suppose to allow IEDs to be exploded at a safe distance by Hv Snipers.
(I always love when a tank 3-4 hexes away from one is destroyed).

Booby traps are generally less effective then mines and are primarily anti-personnel devices, usually more annoying then dangerous for the most part.

Yes, the number of men in an engineer unit matters, the more men in the unit the better it's chance to clear something.

Mines represent a number of explosive weapons, so the status is used to indicate the progress in totally clearing them from a hex. IEDs and Booby Traps are single weapons, so they're either there or they ain't.