In the 17th century a woman only let her hair down to bathe or brush, otherwise she had it pinned up. To have it down for any other reason the woman was considered uninhibited and/or wanton. ( source ) When we use it today its meaning is broaden to include either gender and its meant to act in a free/relaxed manner.

Originating in the sport of boxing this phrase has to do with the countdown a ref gives when one of the boxers hits the floor and can't get back up. In a figurative sense it refers to someone being unable to complete something and is defeated. 

The origin of today's idiom is quite literal. Apparently in the game of marbles you hold them with your knuckles down. So in the game knuckles down means you are playing a serious game. When we say it today we are saying, "Get down to business." ( source )

This expression is believed to have originated from horse racing, where a wire was hung up over the finish line to help determine a winner. Thus, races that were extremely close could be described as "coming down to the wire," quite literally. Read more about it here .  These days we use this idiom when an outcome is unknown until the very last millisecond.