Making progress with the Latitude. First I replaced the CMOS battery and heat shrinked it back up. After setting the date+time (using externally keyboard since the internal one works just sometimes) it no longer errors on boot and holds bios settings.
However it still wouldn't boot to anything and tried to go to the bios nearly every boot. It also randomly kept rapidly beeping like a stuck key, but would stop if I pressed any key on the build in keyboard. It still turns on every time the keyboard is connected and the power button does nothing.
I think I know why. Likely the corrosion is shorting the keyboard matrix and causing it to behave oddly.
Thats when I tried to plug the keyboard in so it would turn on, and then immediately unplug it. And to my surprise, it worked perfectly when I did that.
Bios functioned properly and it even booted perfectly to an XP DVD.
So it looks like the motherboard is fine, but the keyboard is completely messed up and was causing all of the issues before. New keyboard will be ordered soon. And probably some nicer ddr2.