Is it possible to disenchant the Dark Brotherhood armor set's shrouded gloves and put it's enchantment (Double backstab damage) on some other piece of gear? Would this work on shrouded boots' silent movement-enchantment as well?

It's not.

Not the gloves. But you can dischant muffle (silent movement) from other armors you find. Which I find to be a good enchantment for companions when I'm sneaking