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1. What is the LPC, what is it for, and why is it that I can just remove the solder from there or even completely destroy it and there are relatively no consequences?
LPC is a serial data bus. It was probably used in manufacturing for programming a BIOS to the empty TSOP chip on the motherboard. If you damage traces on your motherboard the xbox doesn't need to operate it may still work.
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2. Why does a modchip "boot" from an LPC? I have gathered so far that one of the connections in the chip installation is to the BIOS of the motherboard, but I'm not exactly sure HOW that works or WHY it boots the LPC instead of just causing an error because the regular BIOS can't be booted.
It will only boot from LPC if you ground d0. See #1 for why it does this.
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3. At what point can I consider the LPC "destroyed" in that I literally can't install a modchip? Even if it looks really messy, that doesn't necessarily mean it's damaged beyond repair, right?
Almost anything can be repaired if you have the resources to do so.
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4. Why can't the modchip just be put somewhere else? I suppose this can be answered in points 1 and 2, but just in case.
You could put it anywhere you'd like if it works for you. Near the LPC or attached to a pinheader on the LPC is probably best.
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5. Why are there so many essentially empty "holes" in the motherboard with traces leading from chips to them? I don't understand the purpose of them at all.
They are electrical connections for various digital signals/power. The 'holes' are normally to connect to the other side or layers in the board.