1) The DM is bugged. Until then, you can use noexotics to remove the exotic tag and tame all those critters if you want to.
I dispose of them into the top level of a tower that's several Z high. On the top level, we have a bunch of hatches, ONE pit zone covering all the hatches, and an animal stockpile that does NOT allow empty cages (a separate stockpile for only empty cages draws from this pile to get the empties out of the list). You pit goblins and anything else you want to kill in from here.
Below that is a _retracting_ bridge surrounded with fortifications that they fall onto. Use this for marksdwarf training if you like. Make sure it's inside a big room so that you can station the marksdwarves here (and prevent other dwarves from seeing the trapped creatures and getting scared. Be careful to disarm your prisoners first. You don't want a bowgoblin in here who is still armed.
When you're done killing them, or just bored and want to dispose of them, pull the lever that controls the bridge to drop them down to a lower floor.
My bottom floor is a big atom smasher. Be sure that this bridge is one tile longer than the rest of those in the tower (so that you don't drop anyone on top of the atom smasher bridge. I have a locked door (so that if there are only dead goblins in there, we can retrieve any last goblinite that was missed), followed by a few cage traps (insurance) and finally a floodgate (last line of defense). Mostly I just atom smash the dead at this point. You could instead use spikes on repeat, but it makes it harder to destroy the garbage.
Note that the control levers do NOT have to be near the device. In fact, it's better if their farther away, should a goblin get loose. I think that trolls can destroy bridges, so make them out of something sturdy and kill them quickly (marksdwarves or atom smashing, your pick).
2) Taking their stuff: Yeah, use d-b-d to dump it, but you also have to use d-b-c to claim it (it's all forbidden to start with). And then you want to *undump* the cages (use k if you have only a few, or the stocks screen to do it en masse..... none of your cages should be marked for dumping, so that makes it easy). If you don't undump the cage, I think they just drop the whole cage into the dump, which isn't what you want. You don't have to build the cage anywhere (but if you do, you can skip undumping the cage, because they can't dump a built cage, so they won't).