Mold problem in basement

No, there's no need to install a vapor barrier between two conditioned spaces. The purpose of a vapor barrier is to prevent warm moist air from traveling through the insulation and condensing on the cooler side, which can easily happen in cold climates. With a conditioned space on both sides, you can allow any moisture in the air to pass through.

There's also no need for insulation except for sound proofing (fires can pass through doors, as can any heat). For sound, make sure your insulation is designed for that, and also consider a resilient channel or similar product that will keep the drywall physically separate from the studs, which is how many sounds travel through walls.

Good question. Under the circumstances, I feel you would be better off leaving the existing insulation alone, or simply cover it with a more attractive plastic sheet or some inexpensive fabric. perhaps a curtain type wall over the insulation? If you remove it without a plan to replace the insulation, the basement will be somewhat cooler. There are several ways to insulate the walls without framing, but adhering rigid foam insulation is expensive, up to $80 per 4X8 sheet, and spray on foam is more expensive than that and should be professionally applied.