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I can probably find a cheap dell laptop on ebay. Dell sells power adaptors for cars so finding a power supply shouldn't be hard, am I right?
Yes, you're right.
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I won't need a laptop battery since it'll be connected to the power the whole time.
You won't but it'll make your life much easier. From a CarPC/Laptop user, trust me.
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I am not really a picky person so I was thinking about mounting the laptop under the passenger seat. As for powering the laptop on and off, I could probably extend the wires from the inside of the laptop power button to the outside then I just could reach down hit the power and let the PC boot. I'll do the same when I'll power it off. Is this possible?
Yes, it's possible but it's a pain in the *** to have to do it all the time you get in/out the car. Use WinXP's auto-hibernate when the laptop is on battery to power off. For automatic power-on, use a relay circuit. See this thread.
http://www.mp3car.com/vbulletin/user-innovations/75071-laptop-auto-power-turn-module.html
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I have my eye on a $300 1Ghz laptop... I don't see how this can be more than buying a MB + CPU + HHD + PSU + MEMORY and buidling yourself or am I missing something here?

Don't listen to those PC users who try to scare you off with costs. A laptop can be had for cheaper than an mini-itx setup. You don't even have to buy used. A new Dell laptop can be had for around $500. Look for deals on Fatwallet.com.