ok, i will try to answer some of your questions, but i go off on tangents easily, so you might have better luck searching the forums...
for the case, there is not a huge advantage to using a automotive case other than to keep everything in a small, compact area, though some of the cases offer vibration dampening for certain parts, but the jury is still out to weather it helps or hurts(there are very good arguments either way-- it seems solid mounts are not proven any better than vibration dampened mounts
for most of the boards that require 12 volt input, these require REGULATED 12 volts, so a power supply is still required.
for a power supply, i would highly recomend a opus, or a dsatx, but neither come very cheaply, so might not work. a small board should be able to use a m1/m2/m3 without issues, and can be had for relatively cheaply, just make sure that you get a real m1/m2/m3-there are many fakes on ebay...
the sound card is personal preferance-- for higher quality sound, it is recommended, but with you budget contraints, i might recomend skipping it at first...
for the fan/heat issue, most here have not had too many problems with either heat or cold, most systems can take most of the fluctuation, one of the more common isues i have read is that in extreme lower temps, the grease in laptop drives freezes up, and they must let the hdd warm up to get the grease to flow, and then restart... i live in illinois, and installed my car pc in about july this year, so far, my car sits outside most of the day in the sun, (the temp gauge on my alarm key fob reports temps around 120 deg F), and so far, i have not had any heat related issues with any of my setup(only software problems

)
if your talking about the carnetix p2140, i believe this would be what you need to use with one of the 12 volt input boards.
a invertor is sometimes cheaper, but are usually plagued by noise issues, the bonus to having a m series power supply would be the integrated start up/shutdown controller, that you could use to put the computer into hibernate between all yoru freqent stops.
i hope that answers most of your questions, if you have any more though, try to search first, but you can always post it here. and check out the FAQ area-- lots of good info there.