Basically the latter, eventually, and it wouldn't take very long, the system will reach thermal equilibrium and settle into emitting IR radiation of the appropriate frequency at a rate equal to what the star used to produce, just redshifted down somewhere in the <350 Kelvin region but above presumably 100 K, since the waste heat would need to be lower than what something alive would tolerate.



Sure, but keep in mind something like the Earth gives off a billionth of the illumination that Sol provides, in a mix of reflected light and IR waste. In theory you spot a planet that's on the far side of it's star relative to us by looking for visible light reflection and on the near side relative to us by the IR waste heat the night side emits. In practice we spot them by doppler, transit, etc... I don't think we've direct imaged anything smaller than a Jovian yet, visual or IR. Keep in mind the relative brightness. A planet giving of a billionth of the light of its star, a star we can see at say 300 light years distance, is as visible as that star would be at the square root of the difference in brightness further away... square root of a billion being about 30,000, times 300 Ly, basically the planet's about as visible as the star would be 10 million light years away, and when we talk about seeing stars in other galaxies, we're mostly talking about the really really big ones, solar luminosity rises with the 3rd or 4th power of mass, like 3.5 for main sequence, twice the mass, a dozen times the brightness, thrice the mass nearly 50 times, 10x and you're in the mid-thousands, and a lot of those ones we individually pick out in other galaxies are a million times brighter than Sol. So direct imaging is not usually an option currently, even knowing where to look, plus there are other issues looking at objects very close to stars.