In my reading I find posts recommending warm NP due to spice's preference for it. If this is correct, does this mean that spice that is dissolved in warm NP will "fall out" if the warm NP is left to cool down to room temperature?

Sort of... Depends how saturated it is and how warm it was, some spice might start falling out at room temperature, yes. But not all of it, so to get it all out you need to freeze it.

If you let the saturated pull cool to room temperature slowly (putting in a bath of warm water after its pulled, letting the water cool slowly), then into the fridge once its room temp, then into freezer (and alternating fridge/freezer), you're gonna get some very nice crystals growing.