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.
1906, named for U.S. president Theodore "Teddy" Roosevelt (1858-1919), a noted big-game hunter, whose conservationist fervor inspired a comic illustrated poem in the New York Times of Jan. 7, 1906, about two bears named Teddy, whose names were transferred to two bears presented to the Bronx Zoo that year. The name was picked up by toy dealers in 1907 for a line of "Roosevelt bears" imported from Germany. Meaning "big, lovable person" first attested 1957, from the song popularized by Elvis Presley.
"smoked herring" early 15c. (they turn red when cured), as opposed to white herring "fresh herring." Supposedly used by fugitives to put bloodhounds off their scent (1680s), hence metaphoric sense (1864) of "something used to divert attention from the basic issue;" earlier it simply meant "a false lead":
Though I have not the honour of being one of those sagacious country gentlemen, who have so long vociferated for the American war, who have so long run on the red-herring scent of American taxation before they found out there was no game on foot; (etc.) [Parliamentary speech dated March 20, 1782, reprinted in "Beauties of the British Senate," London, 1786]