mid-14c. (implied in rushing), "to drive back or down," from Anglo-French russher, from Old French ruser "to dodge, repel" (see ruse). Meaning "to do something quickly" is from 1650s; transitive sense of "to hurry up (someone or something)" is from 1850. U.S. Football sense originally was in rugby (1857).
Fraternity/sorority sense is from 1896 (originally it was what the fraternity did to the student); from 1899 as a noun in this sense. Earlier it was a name on U.S. campuses for various tests of strength or athletic skill between freshmen and sophomores as classes (1860).
"plant growing in marshy ground," Old English resc, earlier risc, from Proto-Germanic *rusk- (source also of Middle Low German rusch, Middle High German rusch, German Rausch, West Frisian risk, Dutch rusch), from PIE *rezg- "to plait, weave, wind" (source also of Latin restis "cord, rope").
Old French rusche probably is from a Germanic source. Used for making torches and finger rings, also strewn on floors when visitors arrived; it was attested a type of "something of no value" from c. 1300. See OED for spelling variations.
"a hasty driving forward," late 14c., from rush (v.). Sense of "mass migration of people" (especially to a gold field) is from 1848, American English. Football/rugby sense from 1857. Meaning "surge of pleasure" is from 1960s. Rush hour first recorded 1888. Rush order from 1896.