c. 1300, pushen, "to shove, move onward, strike with a thrusting motion, thrust forcibly against for the purpose of impelling," from Old French poulser (Modern French pousser), from Latin pulsare "to beat, strike, push," frequentative of pellere (past participle pulsus) "to push, drive, beat" (from PIE root *pel- (5) "to thrust, strike, drive").
Transitive meaning "urge, incite, press" is by 1570s; that of "promote, advance or extend by persistence or diligent effort" is from 1714; intransitive sense of "make one's way with force and persistence (against obstacles, etc.)" is by 1718. The meaning "approach a certain age" is from 1937. For palatization of -s-, OED compares brush (n.1); quash. Related: Pushed; pushing.
To push (someone) around "bully, browbeat, domineer" is by 1923. To push (one's) luck is from 1754. To push the envelope in the figurative sense is by late 1980s.
cart n.
c. 1200, "two-wheeled vehicle," usually for one horse and often without springs, from Old Norse kartr or a similar Scandinavian source, akin to and replacing Old English cræt "cart, wagon, chariot," perhaps originally "body of a cart made of wickerwork, hamper" and related to Middle Dutch cratte "woven mat, hamper," Dutch krat "basket," Old English cradol (see cradle (n.)).
Many old allusions are from the cart being used to convey offenders to the gallows (and sometimes serving as a drop for hangings) or for public exposure, especially of lewd women, either in the cart or tied to its tail. Compare tumbrel. To put the cart before the horse in a figurative sense "reverse the natural or proper order of things" is from 1510s in those words; the image in other words dates to mid-14c.: put the plow (sull) before the oxen.