[Kalimac]

If I remember correctly, the only times that hands are specifically referred to is when someone is drawing a sword, and invariably the sword is held in the right hand. I don't have the passages in front of me but I believe some if not all of the hobbits are referred to as holding swords in their right hands; pretty sure there's a reference to Sam and Frodo both holding things in their right hands in Shelob's lair, and when Merry stabs the Witch-King, the sword withers in his right hand I believe (might be getting it mixed up with another reference).

As for Elves, hey, why not? Even if they weren't born lefthanded, when you've got eternity to play with, teaching yourself to be ambidextrous should take a very insignificant amount of time for you indeed.