mid-14c. (implied in rushing), "to drive back or down," from Anglo-French russher, from Old French ruser "to dodge, repel," which is from Latin recusare"make an objection against; decline, refuse, reject; be reluctant to" (see recuse; also compare ruse).
The meaning "do something quickly" is from 1650s, hence "to move or act with undue eagerness or without deliberation or preparation;" the transitive sense of "to hurry up (someone or something), cause to go swiftly" is from 1850. The sporting sense in U.S. football originally was in rugby (1857).
The fraternity/sorority sense is by 1896 (originally it was what the fraternity did to the student); from 1899 as a noun in this sense, "entertainment for prospective pledges." Earlier it was a name on U.S. campuses for tests of strength or athletic skill between freshmen and sophomores as classes (1860).
"plant growing in marshy ground," having leaves that grow as stiff pithy or hollow stalks, Middle English rishe, resh, rosh, rush, etc., from Old English resc (Kentish), risc, rysc, from Proto-Germanic *rusk- (source also of Middle Low German rusch, Middle High German rusch, German Rausch, West Frisian risk, Dutch rusch), perhaps from PIE *rezg- "to plait, weave, wind" (source also of Latin restis "cord, rope"). Old French rusche probably is from a Germanic source.
The remarkable variations in the vowel of this word make its precise history far from clear. [OED]
The stalks were cut and used for various purposes, including making torches and finger rings; they also were strewn on floors as covering or when visitors arrived; it was attested a type of something weak or of no value by early 14c.
"a hasty driving forward, a tumultuous charge," late 14c., from rush (v.). Sense of "mass migration of people" (especially to a gold field) is from 1848, American English, in reference to California. The football/rugby sense is by 1857. The meaning "surge of pleasure" is from 1960s.
Rush hour is recorded by 1888. Rush order, one for goods required in a hurry, is from 1896. The sense in rush of business (1849), etc. is "extreme urgency of affairs."