No... no one is asking questions like that because a chip that consumes no power but still does work would be greater than 100% efficient and would violate the laws of thermodynamics.
Well sure, but exploring the _idea_ of zero power consumption might lead to interesting discoveries in efficiency and power consumption models, even if actually achieving 0% power consumption is impossible.
Yes, if the problem is viewed broadly, eg. passive packages like RFID tags. Power consumption has been a focus area in chips since forever.
For active chips, it's not possible if the problem is viewed narrowly however it's possible to have a package that contains both power generation capability (eg. solar, kinetic) and highly power efficient, sporadic processing which probably exist already. Major application areas would be long-lived deployments (eg. animal tags, probes, implants, wearables).
Is it physically possible to produce a chip that consumes no power? How would that be done? What would be required?
Is anyone asking questions like that?