Thermal energy is primarily dissipated as infrared light which moves at the speed of light. There is no way for space to accumulate heat. If that were the case the entire solar system would be unlivable. The IR emitted by satellites is truly negligible in comparison to the electromagnetic radiation emitted by the sun.

Radiators in space work by radiating electromagnetic energy(light). Heat can only accumulate in matter, not in space, so that is definitely not one of the things we need to worry about.

With radiators just like with every existing satellite system.

https://youtu.be/DCto6UkBJoI&t=12m57s

Very large scale datacenters would likely have some nasty fluid handling problems to solve.

I’ll just note that I am not a fan of putting internet infrastructure in space. I think polluting the upper atmosphere with a bunch of metals every time a satellite deorbits will certainly have negative consequences. So IMO space should be limited to things we can’t do with earthbound infrastructure.