okay so you want to put bottoming cycle thermal powerplant on waste heat? am i getting that right?

so now some of that heat is downgraded to lower temperature waste heat, which means you need bigger radiator. you get some extra power, but it'd be a miracle if it's anything over 20%. also you need to carry big heat engine up there, and all the time you still have to disperse the same power because it gets put back into the same server racks. this is all conditional on how cold can you keep condenser, but it's pointless for a different reason

you're not limited by input power (that much), you're more limited by launch mass and for kilogram more solar panels will get you more power than heat engine + extra radiators. also this introduces lots of moving parts because it'd be stirling engine or something like that. also all that expensive silicon runs hot because otherwise you get dogshit efficiency, and that's probably not extra optimal for reliability. also you can probably get away with moving heat around with heat pipes, no moving parts involved

also you lost me there:

pull in cold dry air, add water to cool it further

okay this works because water evaporates, cooling down air. this is what every cooling tower does

make it more conductive

no it doesn't (but it doesn't actually matter)

condition again to recover water and energy

and here you lost me. i don't think you can recover water from there at all, and i don't understand where temperature difference comes from. even if there's any, it'd be tiny and amount of energy recoverable would be purely ornamental. if i get it right, it's just hot wet air being dumped outside, unless somehow server room runs at temperatures below ambient

normally radiation only works because the radiator is more conductive than the rest of the system, and so it tends to pull heat from other components.

also i'm pretty sure that's not how it works at all, where did you get it from