when you have a dry suit as well, do you tap it off the DIL cylinder, or run another cylinder
Matter of redundancy and used CCR type. On CC you have 3 factors influencing your buoyancy- Drysuit, Wing and LoopVolume. In a perfect world the DIL would only act on the loop. Most CCR have integrated wings connected to the DIL circuit as you wrote and it is in fact the tool of choice for buoyancy control. Drysuits are commonly connected via either Argon bottle seperately or simply hooked to any offboard gas connection.
So far, I have not seen any drysuit directly on the DIL (which does not mean there may not be anybody out there doing it so).
If it comes to DIL usage, (and again a perfect world example) you would ideally only need the loop volume multiplied with the depth plus a bit of flushing- the overall demand is therefore rather low. But its as well a matter of training. In this regard OC differs a bit from CC but I believe general diving experience (no matter if CC or OC) is key to success.
Its perhaps an easier way to see any CCR rather to be some sort of on-the-fly-gasblender. Taking pure eCCR out of the equation it is again a requirement to plan your dive properly and add redundancy in orderly fashion. This of course involves the gas needed for buoyancy.
@JonnyC compared spelling words with knowing the alphabet. It makes sense in a way that critical tasks like stagehandling (Lean-to-Rich, Proper labels, Stuffing Regs etc.) is the alphabet. Breathing a rich bailout on bottom is equal to breathing deco gas down there. No difference. Same a potentially fatal mistake.
I have often been laughed at when pulling out my ducttape and additionally labelling the MOD on the regulators outflows. (Why would you do this? Noone needs that! You only paint your face...) True, I may not need it. But if in stress it may avoid an error. So everyone has additional strategies to create backups of his own brain malfunctioning. And this may happen definitely more likely on CC than on OC.