On radiators, plugging it into this formula:
https://projectrho.com/public_html/rocket/heatrad.php
I get a circular radiator at least a kilometer wide, assuming the radiator is quite efficient, a rather modest datacenter, and very hot coolant (70C).
…Realistically, the coolant temperature would need to be much lower. See how it’s a power of four in the formula? That means the radiator area gets very large real quick.
I cannot emphasize how expensive a functional 1km+ radiator would be in space. It’s mind bogglingly expensive.
If a space datacenter is in LEO like Starlink, then it’s in Earth's shadow a lot of the time, and would have to be “part” of the starlink network constantly zooming over the ground. If it’s geosynchronous, then laser communication (or any communication) gets real tricky, and latency is limited by the speed of light. I’m not saying it’s impossible, but reliable high data rates would be an expensive engineering challenge.