Open the (O)rders menu, the (r)efuse orders submenu, and turn on "dwarves collect refuse from outside." Then, your dwarves will collect corpses from outside, bring them to your refuse stockpile that you have set up, and, assuming your butcher's shop is somewhere near said refuse stockpile and the animal is not yet rotten, the butcher's shop will autoquene a butcher dead animal job. Then, the dead animal will be quened. Hunters will drag the hunted animal back without outside refuse gathering, but for military hunting, you need to do it this way.
There's a few ways to set wild animals up for butchering. The first is to build Kennels and tame them, then designate them for butchering like you would any other tame animal. The second is to release the animal, kill it manually (many people do this by "releasing" it over a deep pit), and then let your butcher go to work. To release an animal, there's a couple of ways of doing it. The first way is to designate a zone (i) over open space, designate it as a (p)it/pond, and set (P)it/pond information such that the desired meat is getting tossed into the pit. A dwarf will go to the cage, collect the animal, drag it to the pit, and toss it in. If the pit is tall enough, the animal will then promptly splat on the bottom, otherwise you'll need some other means of murdering it. Don't do this with dangerous creatures, as they might escape while being dragged to the pit! The other way of releasing an animal is to (b)uild the cage (j) - and note that this does not mean build a cage trap! - that contains the animal, link a lever or pressure plate to the cage, and then pull the lever. You'll have to find another way of murdering the animal at this point. Or, for safety, you can pseudo-combine the two; build the cage near the pit, then set the animal to be tossed in. This makes it less likely that the animal will break free of the dwarf dragging it to the pit, as it has less (or no) time to break free.
By the way, (q)uerying the cages while they're on the stockpile isn't working because the cages aren't buildings until they're (b)uilt. Again, (b)uilding a cage (j) is different from (b)uilding a trap that's a cage trap!
Hunters generally aren't very efficient. It's much better, easier, and safer to simply use domestic animals for your meat purposes; embark with a couple of female dogs and a male or a cow and bull or a male horse and a female horse next time, or buy them from caravans. Once you've got breeders up, you can slaughter all the males and let the females mature so they become more breeders.
Finished goods that are also artifacts are mostly useless. They don't do anything. There are mods that fix this, but except for artifact ropes and chains, you can't actually use them for anything.