Boxing (pugilism) was codified by the Greeks in the Olympics
around 688 BC.
Modern boxing rules date to the 18th century, and the Marquess
of Queensberry rules (including the first gloves) date to 1867.
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Fisfticuffs would be used in bare-knuckle boxing, which is a
variation of boxing enjoyed by some people. The boxer would have no
padding or gloves on his or her hands.
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Down-and-out is from a situation where a beaten prizefighter is knocked unconcious. It originated in 1889 in American English from pugilism (boxing or fist fighting).
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No, it is not an adverb. The word box can be a noun for an
actual or metaphorical container, or a verb meaning to put into
boxes, or to engage in pugilism (boxing, sparring).