It'd be nice But if it is induction charging, that's just charging via magnetic field. Nothing to do with static electricity at all. Which'd probably make more sense, since there's not a lot of static electricity in the environment usually.
Yup. In "The Daleks," their static-electric power source made them totally dependent on staying within their city, and they shut down when it was sabotaged. Just the opposite of letting them operate indefinitely. When they made their second appearance in "The Dalek Invasion of Earth," they were equipped with parabolic dishes that were explained as receivers for a different, broadcast power source, and later models starting with "The Chase" gained the now-iconic side panels that were meant by designer Ray Cusick to be solar collectors. In "The Power of the Daleks," there was actually dialogue about how using static electricity as a power source was believed impossible by human science and must be the product of some extraordinarily advanced technology -- which I take as David Whitaker hanging a lampshade on Terry Nation's bad science. And holy cow, I just realized. A major plot point in that serial is about the Daleks trying to build their own independent power source so they can turn on the colonists, and the Doctor defeats them by sabotaging their power source. So it literally is about "The Power of the Daleks"! How have I seen that story three times in the past year or two, and read it several earlier times in my life, and never figured that out before? Although the Daleks in that serial had the solar collector panels, which makes their dependence on outside power sources anomalous.
would make sense that someone would have lifted it too... we do know that in "The last outpost" the communicator does have gold in it. line: "It is gold". Also not really too relevant, but in 20,000 leagues under the sea, the "nautilus" is powered by the ocean like a wet cell battery. Not really sure why that would work.