Is it harmful for the camera and/or chip to run it without the cooling running?
In a DSLR the chip warms up fairly quick at no harm.
Nope, just means more noise than you would have with cooling running.
What is actually being powered by the 12V connection?
Only the cooling mechanism, or also the build-in USB-hub?
In that case I would not need to place an extra powered USB-hub at the scope end.
Only the cooling mechanism is powered by the 12V DC power adapter. The camera is powered over USB using around 400ma but it could use more (like 600ma) if using the USB hub on the camera as well.
What amp rating do you advise for the 12V feed?
Should this be a clean feed, or can I combined it with the anti-dew heating and/or mount power feed?
At least 3A is recommended. You can combine the power feed with other devices just ensure whatever your source is that it has enough amperage to supply everything connected to it. Remember here that the power is a draw consumption meaning you could hook even a 12V 5A adapter to it and it won't harm the cooling module. It is not going to deliver 5A to the cooling module, it will only deliver what the cooling module desires (sometimes up to 3A). I am stating this because this question comes up a lot, people think that when you hook a 12V 5A, etc. source to the camera it is going to deliver that and fry the camera cooling module, this is just not the case. Ensure 3A and you will be good.