My farther and I are considering setting up a PVR, as it could more features and is cheaper then a HD DVR. We will be using it to record HD Satellite TV and public broadcast HDTV.
Though we have experience in computers and electronics, we have not messed with media much though and we have some questions.
Welcome to the club. It's a fun place to be.

First, we need to know how powerful of a computer we need. We'll planning on using a Linux-based OS for it and that we plan on it recording and playing back HD TV. The hardware we have is out of date that we plan on using, I am pretty sure it has a crippled 2.7ghz AMD with about 700MB or so of RAM(maybe less). Is there enough power here to do this?
For recording things the computer does not need to be powerful at all. What you have should be fine. For playback of SD video the requirements aren't that bad either. However, what does take some horsepower is playback of HD video. And for this your system would be a little on the low side. However, whether you go with this system or build a new one you would have to buy a tuner. So I would say buy your tuner and try it with this system. If it works great. If not, you can build a new one.
Second what Linux operating system would you recommend? The reason for a Linux based one such as Ubuntu, is that they are free. Spending $100+ on a OS alone would shrink out budget for this and it doesn't make to much sense when you can just use a free one.
For linux the best choice for pvr software is MythTV. I don't use linux, but I believe you can actually get a linux build that has Myth integrated already.
Third, do you need a decent GPU for HD Video Playback? I am doubtful you do, as far as I know GPUs' are only really used for gaming or real-time 3D programs, and only process textures, polygons, vertices, lighting, etc. And as none of that is required for videos, we should be fine with an on-board out-of-date GPU correct?
Actually, a good videocard is what might make this system work. You need it for HD playback as most good videocards can unload that to the gpu and save your cpu. Your trick will be finding a videocard that will work with your motherboard and also has linux drivers. I would also buy it local in case you decide to buy a new system you will probably want to return the videocard for one that would work with the new system.
Forth, VGA is fully capable of providing HD Video right? I am pretty sure it is, as my computer uses VGA and the screen is set at 1680X1050. Pretty sure the only problems with VGA is that it is converted to analog, which is supposedly is more susceptible to interference.
True. But this will depend more on your tv and what kind of inputs you have available. What kind of inputs does your 47" have?
One other point to consider is that if you want to record HD from a box then you need an HD analog tuner. There used to be only one choice, the HDPVR from Hauppauge. It works great, however some competition was just introduced. I don't know what kind of support it has yet, but for either tuner you woulld want to make sure there are linux drivers for them. Read more here:
http://forum.byopvr.com/dvr/index.php/topic,10869.0.htmlSo the only other thing you need to find out is what kind of videocard slots are on that old motherboard and then we can try to work all of this out.