Judy Parkinson's From Hue & Cry to Humble Pie tells me this about the expression to fiddle while Rome burns: the expression, which means to do nothing in a crisis, is a reference to the behavior of one Nero Claudius Caesar during, appropriately, the burning of Rome. Apparently, Nero sat up in a tower, playing a lyre and reciting his own poetry while the fire raged. Which isn't fiddling at all. They lied to me through cliche. Uncool, man, uncool.
I wasn't able to determine how fiddle came to mean mess around with, although if I had to guess, I'd say it comes from that. Although it could also be a shortening of the 16th century expression fiddle faddle - faddle is an obsolete term that meant trifle. Or it could just refer to the thing you do with your hands when you're playing the fiddle.
You know the difference between a fiddle and a violin? Pretty much nothing. According to dictionary.com, a fiddle can be any member of the viol or violin family, but it's most commonly just a violin. The distinction comes from the tradition the player's playing in.
Know what else? Fiddle comes to us by way of German, and they're not totally sure, but it might go way back to Vitula, Roman goddess of joy and victory.
Violin is a diminutive of viola, which comes to us by way of Latin and might go way back to... wait for it... Vitula, Roman goddess of joy and victory.