I've heard that you can autokill mobs spawned by soulcages and still get drops (blazes normally drop rods only when killed by hand, if you autokill them when they've come from a cage, they still drop rods). Does this work with wither skulls?
Yes, it does. Unless it's been fixed, you can collect regular skeleton souls, and then place the skeleton shard spawner in the nether to get wither skeletons.