Black plastic is impermeable. If you're going to be excavating the area and then filling it with stone, remember that if the soils immediately adjacent to the area you excavate are anything but really free-draining, the stone filled area may tend to hold water when it rains for. Whether or not that's better than what you have now depends on how fast the water drains away. And if it's in a fully shaded area, mildew and molds and other niceties may become problematic.

Generally, water drains though most landscape fabrics. So, if you must line the area with something, go with fabric that is permeable.

I agree with you. That black plastic is going to show up anyway and as dirt collects between the crevices of the rock you'll get your unwanted weeds anyway growing on top of the plastic (plus no drainage). I would definitely just put it on dirt. Landscape fabric and plastic are worthless. I don't think the mushrooms will be a problem with rock as they develop on decaying organic material, not rock (if you put down bark mulch that might invite them). As for weeds you can either pull them as they come or apply a preemergent in the late winter/ early spring .