<< From reading around here the past few days, I've come to the conclusion that I would be cool with a Athlon XP 1600(seems to have the right balance between performance and price). >>
I personall would make the jump to the 1700+ because it doesn't cost that much more and gets more performance. The 1800+ and 1900+ are still priced a bit high, IMO.
<< The main things I'm wondering about are: what steps should I take to keep the processor cooled well (I will not be overclocking at all)? >>
If you will not be overclocking at all, then get a retail boxed processor that comes with a fan and thermal tape already applied. You just have to install the processor and HSF on the motherboard and you are ready to go.
<< what is a good, stable motherboard? >>
I think the SiS 735 chipset on the ECS K7S5A is good or the VIA KT266A on the Shuttle AK31A (and many others) is good as well. These are basic boards at good prices, there are many other options out there, so you'll need to look around.
<< what kind of case/PS should I get? >>
I like the Enlight 7237 series and the basic 300W PSU that comes with the normal one should be okay for most things, but if you want a little extra power, pay the extra $20 or so and get the one with the 340W PSU.
<< and what's the best value for a video card right now? >>
Since you don't do a lot of gaming, then consider the Radeon or Radeon VE from ATI which is in the $50-60 range and has really good 2D for your office apps and such. If you like NVIDIA and want to keep the costs down, the MX200 is available for a little less, but the 2D is not quite as good in some cases. If you are willing to spend more, there are lots of options out there above this range, so you'll need to look around.