If I would like to put the noisy computer case away from my desk, in
another room for instance, how would I go about that ?
If it's really heavy, try a forklift.
If it's really light, put wheels on it, a sail on top, and
blow really hard.
How long may video, mouse and keyboard cables be ?
Using the cables you already have would be an ideal length.
Would I have to use some kind of amplifiers ?
Only if you want your music louder.
Now to be more serious-
You can't just do that, extend cables and up the signal
voltage. It degrades the quality to the point where it's
non-usable for typical tasks, far far less desirable than
putting up with the noise.
HOWEVER,
This is all ridiculous. There is no reason ANY PC needs to
be noisey. Address the problem directly, optimize the
cooling for low noise. That does not mean it needs be
expensive, or water-cooled, or any of this. Just use the
time-proven methods millions of people and OEMs have used
for years. A noisey system is, IMO, a maldesigned system.
Fix the problem rather than moving the problem to another
room.