Some tips for better separation:
After letting your lye/root bark solution soak for a while (I think Noman's says for an hour so), add some more lye right before putting in your naphtha. This will heat up your solution, which I've found helps the naphtha separate not only faster, but more completely. You could just as easily keep your container in a hot water bath (careful if it's glass; heating/cooling can fracture it), but the potential pH boost from the additional lye will likely also help for separation.
Also, let your naphtha sit in a hot water bath (obviously away from any open heat source) to heat it before pouring it into the solution. Warm naphtha will separate more easily, and the warmer it is, the less you will have to use to saturate it, which will help with freeze precipitation This will also allow it to pull out a lot more impurities than colder naphtha, but if you're following Noman's recrystallization steps that shouldn't be an issue.
Oh, try to use a container that is as skinny as possible, this will make it easier to siphon off the solvent layer.
As far as discarding the lye solution, lye very common in drain cleaners, so you'd probably end up with cleaner pipes by pouring it down the drain
However, I don't know enough about specific materials in pipes and chemical reactions to safely advise doing this, so it might be better just to neutralize it with vinegar first or compost it as others have said