Basement Finishing Question (insulation)
You shouldn't remove the insulation in your basement's walls unless you plan to replace it. Even if you're furring out the existing walls with new full-height walls for the finished basement, it is critical that the outer "downwall" gets insulated.
I think I'd go for one vapor barrier only, and I'd want it as close to the conditioned space as possible. The paper vapor barrier face comes off most batt insulation fairly easily.
It won't be the end of the world if you add another vapor barrier. There are pros and cons to doing that, but I don't think it will be a big deal either way.