Concrete – Do I need a vapor barrier on an 8″ pour + carpeting over tile

Pull out the carpets. never put a subfloor over something like old carpet. You may not have any moisture problems now, but any occurrence of water (flood, broken plumbing, spill) getting in that hidden carpet in the future will cause you more grief than you could dream of. This would be a perfect situation to grow mold, mildew, stink and rot the wood subfloor from the bottom up. Don't take the chance, remove it before putting down anything else.

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.