It doesn't, but whatever helps you remember. :)
Chauffeur comes from the French 'chauffer', 'to heat', and was originally used of steam engine stokers; people tending to the furnace and thus driving the boat forward. We have the word 'sjåfør' for driver in Norwegian as well.
Kjøre comes from the Old Norse 'keyra', meaning to ride, drive forward, set in motion, thrust, or spur something on.
I would have thought so, but that doesn't seem to be the case.
"å kjøre" comes from the Old Norse "keyra", with roots all the way back to the proto-Germanic "kaurijaną" (reconstructed).
"scheuren" on the other hand seems to take its meaning - and etymology - from the other meaning of the verb, "to tear"; so you get something like "to drive so fast your car starts tearing apart" or "to drive so fast you tear up the road", I suppose. Your guess is probably better than mine on the Dutch front, but that's the link Wiktionary gives. :)