I want to build a new desktop computer on which I plan to install
Linux.  My priorities are:
- strong linux compatibility, including ability to do TV out
- quiet...would possibly consider a micro form factor but not necessary
- I have a few parallel IDE-based drives that I would like to throw
in, so having 2 PIDE channels is a bonus (though not necessary).

Please describe some pros and cons about various CPUs,
motherboards/chipsets, video cards, and cases that match these
criteria.  I'd expect to pay about $500 total for these 4 components. 
Please describe what I should expect to work or not work in Linux with
each component (particularly the video card).  For all 4 types of
components, what would be your top choice, and why do you prefer that?

unless your incredibly good with linux, id strongly recommend you stay
well away from ATI graphics cards. The drivers for these for linux are
mind bogglingly hard to install, with no simple set up program.
instead you are given a program which must be run through command
line. i tried several times to install mine and eventually gave up.
otherwise, i think just about anything works alright :)