|If I replace the motherboard and video card on a dell, and use the same hard drive, will Windows work normally when I boot it up?|
No, not unless you replace it with an identical motherboard and video card. You will have to do a repair install in order to get the correct hardware drivers loaded.
Also, if I try to re-install the Windows that came with the dell with the new motherboard, will that work? How does Windows verify that it's the original Dell system?
Probably not unless you replace the motherboard with one from Dell. It knows becasue of the BIOS code.
Different rules apply to OEM versions of Windows and Retail versions. OEM versions are tied to the computer it came with. Retail version can be moved to another computer but it will have to be scrubbed on the old computer.