If I take an electrolytic capacitor, remove the aluminium casing, unroll the foil/ electrolyte-soaked paper, if charged, will it still work? And what if I took of the cathode foil and wrapped the anode foil with the paper around a metal rod -- will it work if charged? If not, why?
Dodo, I'm not really sure why you would want to do this.
I do not think it will work when unrolled. This is also dangerous and not advisable. IF you will go through with it anyway, wear gloves and do it in a well ventilated area. If you remove the case and unroll it, it is just a flat piece of foil. The foil doesn't make the capacitor as much as it is the foil AND the geometry. If the anode foil and cathode foil were still ordered properly after removal, it would still work obviously, but I do not think it would work to the manufactories specifications.
If the rod were in direct contact with the layer you put on the rod, yes it would work. To what end you would want that for, I can only guess. Cheers.