How to know whether the default coating on the iron cookware has totally come off?

If all the seasoning is completely removed, it will look shiny like brushed or polished metal. This takes a lot of time, and often power tools. I would first ask whether this is what you want to do: does the pan feel waxy, and have a surface that turns glossy when heated? Then there is wax which can be easily removed with a scrub-brush while water is boiling in the pan. Finish with a little soap and hot water, and you're ready to season.

If it is, in fact, the seasoning you want to remove (if, for example, it is coming off in places, or parts are rusting, typically on cookware which has been left outside), the easiest procedure is to heat the pan very hot, either in the coals of a campfire, or in an oven on 'clean' to burn off the previous coating (don't do this with a pan coated in wax, the wax will burn and smoke), then brush with steel wool until the metal pan starts to show through. You will need to season with several coats of an oil like crisco or coconut, and this will take a lot of time. It's rarely necessary to strip a pan unless it is in awful condition.

At the point you're at, I'd assume that any wax or residue is gone from your work with the steel wool, and that you don't need to go as far as to strip the whole thing. To salvage it, put aside the steel wool, scrub with a plastic brush and hot water until the black residue is gone. Clean a little more using a rag and oil (which will pick up oil-soluble debris) until that is clear. Then season as you normally would, using a light oil like crisco in the oven.