I'm more interested in the practical options then the funny ones.
So...I take it you want both, but order matters? I can do that.
More practical options:
Hunters always "return kill" to a butcher shop, not a food stockpile, so, if you want them to get back to hunting faster, build a butcher shop where you want them to drop their kill, queue a "butcher animal" job on it, a "slaughter animal" job on it, and suspend both. That will prevent the shop from being used for auto-butchery, so it'll only be used by your hunter for dropping off kills. (Note: haven't tested to make sure it works like this, ‼science‼ needed, but there's a decent chance it'll work.)
A hauler will then come and move it to a refuse stockpile.
In order to get dwarves to dump stuff down a hole, you use a garbage zone, not a refuse stockpile. The difference is, a zone can be declared over open space, but a stockpile can only be declared on solid ground, when you declare a garbage zone with one solid square to stand on and one square of open space, dwarves will stand on the solid ground, place things into open space, and you get them automatically dropped off the edge.
Dumping requires manually marking the items to be dumped, so it's not very practical for hunting, as you'd have to micro-manage it.
However, you could set up a stockpile on top of a hatch cover. Haulers will move the bodies from the butcher's shop where the hunter left it to the stockpile. By connecting the hatch cover to a lever or a pressure plate that dwarves trigger when they go to place a body on the stockpile, using an appropriate delay, you can have the hatch cover open to drop the corpse down after the dwarf has left. You'll want to (d)esignate restricted (T)raffic over the hatch-cover, to reduce the chances that anyone will be standing on it when it opens.
The funny ones:
You could instead set it up like this: build the butcher's shop with suspended jobs, as suggested above, but then build a wall around it, force access to it to be from a long, narrow hallway directly above where you want the kills to be dumped. Channel out the ground from this hallway, but leave one square of solid ground at the entrance of the hallway.
Build a retracting bridge over this hallway, and a pressure plate in front of it. Connect the pressure plate to the bridge.
Now, when your hunter goes to drop the kill off at the butcher's, the bridge will retract, and they'll instead drop it off wherever they land.