1650s, "public department in charge of letter-carrying," from post (n.3) + office. Meaning "building where postal business is carried on, office or place where letters are received for transmission," is from 1650s. In slang or euphemistic sense of "a sexual game" it refers to an actual parlor game first attested early 1850s in which pretend "letters" were paid for by kisses.
sink to wash food, dishes, etc., 1824. Phrase everything but (or and) the kitchen sink is attested from 1944, from World War II armed forces slang, in reference to intense bombardment.
Out for blood, our Navy throws everything but the kitchen sink at Jap vessels, warships and transports alike. [Shell fuel advertisement, Life magazine, Jan. 24, 1944]
Earlier was everything but the kitchen stove (1919).