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What would a computer the size of the Death Star need to run effectively?

Imagine: A Kardashev Type I civilization mainly resides on a planet analogous to Earth, with one moon, analogous to Earth's moon. Eventually, the society grows so complex that machines filling space becomes a serious problem: the world's growing population needs food more than it needs machinery, but it still wants to balance both with the space it has.

Their solution: Building a Death Star like object, roughly the size of their existing moon, in orbit. Its function will be doing all computations required for the civilization, and then some; it will handle all storage of data, and other things that the planet can access wirelessly, to preserve space for farmland and residential areas at home.

I am not concerned about the following:

  • How such an object will be constructed. This civilization is advanced enough to build on a lunar or planetary scale, but it just doesn't have space to build at home.
  • Where the resources to build it will come from. Assume that all necessary materials can be procured and transported from the inner asteroid belt with relative ease.
  • Why it needs to be so large. Assume this civilization plans to rely on this object for centuries or millennium to come, so they are building it with far more processing power than they currently need.

However, I need specific answers as to how this satellite will do the following:

A) Orbit without colliding with the existing moon or flying off into space. Assume that much of the space inside the satellite is non-flammable gas or vacuum, so it doesn't have enough gravity to cause imbalance within the system.

B) Power itself. Solar power can not be used as only 50% of the surface area of this sphere will ever receive sunlight at a time, and most of it will be at an angle too acute or too obtuse to use effectively with stationary solar panels. Furthermore, less than half of the satellite's surface area would then need to power its entire internal structure, which is not viable.

C) Maintain a healthy internal temperature at which machines run efficiently. This must take into account both how means of generating power will effect temperature and how radiation from space will affect temperature.

D) Effectively maintain and repair itself like a living organism. This is not mandatory but it would help keep the satellite running smoothly.

E) Remain structurally sound. While this object's gravity will be weak, so the local system will not be altered, its innards should not be crushed from its own immense weight.

What reasonable scenario(s) for each subquestion would allow this object to function?

If possible answer in A), B), etc format.

Note: This is my first question. Please let me know if I should change the format in which I ask questions, if these should be separate posts, if the tags are inaccurate, etc. All feedback is greatly appreciated.