First, Welcome to mp3car.com forums!
Second, before you build your computer, ask yourself, what will I use my carputer for? What is seems in your setup, you could play call of duty, flight simulator 2004, top of the line games. I know some people want to have the best of the best, but if I was you, I will rethink the whole carputer and make a good plan.
My recommendation are here: Start with paper and pencil (or pen). Write down what you will use your carputer for. For example, GPS, TV, radio, MP3, DVD, games, DIVx, Internet access...etc...the list keeps going....Then using your computer knowledge, ask yourself whatever the purpose of your carputer, what parts will be good for it.
If you still plan to use the parts you listed, it is very nice, not a bad choice. However, keep in mind that size is a factor, I know you will be putting it in the trunk, using a normal sized tower, I don't know. Someday you might need the space in your trunk to load something up and that case will be on your way. Why not use a mini-atx board and case? Since you will only be using un AGP slot and maybe 2-3 PCI, why get a normal size tower?
About the power supply, there are different choices, like many people say here, the best so far out there is OPUS...you can visit their website at www.opussolutions.com -They don't sell it off the web, you need to email the guy in order to get the price. These power supply are special and have advantages over other, however the price of these PSU are from $150-$200 depending if you get 90W or 150W. The biggest one is the 150W, it may sound small the number, but you can do a lot with the 150W. You can even power a P4 system. What are the advantages....you can check the forums for that cause there are sections on Power Supplies and all about the OPUS.
But a quick listing, it has the shutdown controller (it will shutdown your carputer after xxx minutes after you turn the engine off) The carputer will not reboot when you crank your car, and it will boot your computer automatically when you turn the switch on, you can also put a switch instead of wiring it to your ignition. There are other features but this is the best feature it has.
The idea of the inverter is the cheap and easy way to it. However, if you stop for gas, you will need to turn your computer off, wait for windows to shutdown, get gas, then when you get in, turn it back on...etc... Unless you wire the inverter directly to the battery, however it may draink your battery someday or shorten your battery life. This might work, but with an OPUS, its hassle free. Maybe its ok to turn the power off without shutting down your computer once in a while, but you might kill your HD, Windows system files someday. For some reason, Windows, Mac, Linux, etc...may a option called "Shutdown". If nothing will happen, like some people say to power off the mean way, then I don't think we will ever need that "Shutdown" menu.
Yes you can run your PC with the car off but not too long. First it depends how much stuff is running and how much power it is using. a 350W PSU, 120GB HD, RADEON 9000 Video card, AMD XP 2600, that is a lot of power that you will be using. I don't recommend it unless you have a seperate battery, but you will need to charge it when the car is on. Making a switch in the front sounds easy and is easy, you need to find a switch and a longg wire. By the way, I recommend an external USB Dvd rom drive, it won't make sense to have an internal one in your trunk and have to stop your car or have someone go to the trunk to change a disc.
And you are missing the most important piece. Monitor, how are you planning to control your computer? Using touchscreen? What size are you planning to get? How are you planning to install the LCD screen? Go to the section of LCD Display and you will learn a lot from there. Most people get the 7" Lilliput or Xenarc, because they have VGA input, Composite is not as good as VGA in quality sense, so think and make a good plan before you start building.
Thats all, hope this helps.