I don't really want a laptop cuz I can get much more for my money with a
desktop. Is there anyway to hookup a keyboard, monitor and mouse to a
box that is in another room, such as in a basement, so I can save space
in a bedroom, and not have to listen to the fan, etc?
No, it cannot be done reasonably, especially considering
that your desire is the desktop for the cost savings over
the laptop.
A desktop system need not be large though, especially if you
can make due with integrated features such as sound and
video, some of SFF systems are indeed small form factors.
As for listening to the fan, what are you comparing to?
Taking a random generic OEM system of modest configuration,
a Dell for example, is not very loud if you don't buy the
high-end, high-heat CPU. Going with an HP or Compaq that
uses an AMD64 CPU should then help even more since they
produce less heat. Even on my overclocked gaming system,
fan noise is only audible if the background noise is very
low but for that it was of benefit to have a larger case,
thus room for larger lower RPM fans. It would not be hard
to buy a new system that produces less than half the heat
and so cooling is easier, quieter.
More than anything, trying to string long cables you will
find a degradation in video quality. Even using extenders,
the quality suffers after it goes past a few (single-digit)
feet of cabling. Digital has timing issues and analog even
worse at modern high resolutions and refresh rates. Your
best bet is to just buy what you described, a small quiet
system and use it the traditional way. If you can't find
one with acceptibly low noise levels you might then look at
installing a better heatsink (so that fan can run slower),
or in some cases (pun intended), some modifications to
improve intake and exhaust flow so the fans run at lower RPM
to attain acceptible flow rate.