Could a habitable planet lit by the cosmic microwave background plausibly exist?
I've got this idea of a rogue planet that is moving that fast through the intergalactic void that the cosmic microwave radiation, thanks to the Doppler effect, actually provides enough heat that it is habitable (and of course, it should be actually inhabited). Thanks to the searchlight effect, I think the incoming radiation should even look roughly star-like, though I haven't done any actual calculation on this.
However I wonder how plausible such a planet would be. In particular,
Is there any plausible (though possibly very unlikely) way such a planet could come into existence?
Is it plausible for that planet to exist long enough in this state to develop life? After all, eventually it will have to interact with something else (though I aimed at minimising that by having it pass through an intergalactic void).
I'd actually also like to have a moon orbiting that planet. Is it plausible for such a planet to have obtained one?