1.) Yes, you can. Simply mark it all for dumping. An easy way is to press d -> b -> d over the cage, designating it like mining, and then k to look at it and find the cage, and unmark that, since you don't want the cage dumped. Dwarves should trot up and strip the gobbos of their equipment. (I think, I've never done it, lol. Also make sure to have a garbage zone defined so they'll actually dump it, and not into lava or something.)
2.) What kind of artifact is it? Most artifacts will be placed in the stockpile of the appropriate type. If it was clothing, someone might be wearing it. If you give your dwarves cabinets, you can also check those to see if it's been stored somewhere. An exception is gem artifacts, they won't go to any stockpile for some reason, but just sit in the workshop. Press t over the workshop it was made in to see if it's still there - a dwarf might not have hauled it yet.
3.) Have you assigned the burrow you want them in to the alert too? If you haven't linked it through the military screen, they won't go. Also try not letting them go outside with the Orders screen (press o)
4.) Nest boxes are claimed automatically, but egg layers don't lay constantly. They also leave the box early if a dwarf collects the eggs. Just keep watching the boxes, and if you want the eggs to hatch, be quick to forbid the eggs before the dwarves take them. (Also make sure you have females, just in case).