Second one first, yes connect the pump to the CPU fan header as you did.
The description you gave of the quandary about the case cooling is the exact reason I went with the Obsidian series (in addition to the case being a bit bigger). But I wound up using a double thick radiator AIO that has push-pull fans in series instead of side by side blow like you have. You could probably go that way in your case too, allowing you to push the air
out the back of the case instead of pulling the air
into the bottom of the case, warming it up with the CPU heat, and then spreading that on the other components. It would probably cost you less to buy that new cooler than to buy a bigger case with better cooling characteristics. I'm not sure if Nzxt makes one in that configuration, I know Corsair does.
Davemon50