Middle English rop, from Old English rap "strong, heavy cord of considerable thickness," from Proto-Germanic *raipaz (source also of Old Norse reip, West Frisian reap, Middle Dutch, Dutch reep "rope," Old Frisian silrap "shoe-thong," Gothic skauda-raip "shoe-lace," Old High German, German reif "ring, hoop"). Technically, only cordage above one inch in circumference and below 10 (bigger-around than that is a cable). Nautical use varies. Finnish raippa "hoop, rope, twig" is a Germanic loan-word.
It is attested by early 14c. as "a noose, a snare." Rope of sand (1620s) was an old figure for anything lacking coherence or binding power.
To know the ropes "understand the way to do something" (1840, Dana) originally was a seaman's term. The phrase on the ropes "about to be defeated" is attested from 1924, a figurative extension from the fight ring, where being in or on the ropes was a figure by 1829.
To be at the end of (one's) rope "out of resources and options" is attested by 1680s. An earlier expression was have too long rope "have too much freedom" (late 15c.).
Rope formerly also figured in slang and extended-sense expressions related to punishment by hanging, such as John Roper's window "a noose," rope-ripe "deserving to be hanged," both 16c. The figurative phrase give someone (enough) rope (to hang himself) is by 1680s.
mid-15c. (earlier ren, late 14c.), "a spell of running, the act of running," from run (v.).
The Old English noun ryne/yrn (early Middle English rine) meant "a flowing, a course, a watercourse;" the modern sense of "small stream" is recorded from 1580s, mostly in Northern English dialect and American English. The sense of "a flowing or pouring, as of liquid" is by 1814. In reference to the action of a school of fish moving together, especially upstream or in-shore, by 1820.
From 1804 as "place where anything runs or may run." The meaning "the privilege of going through or over, free access" is from 1755. In. U.S. baseball, "feat of running around the bases without being put out" by 1856; the sense in cricket is from 1746.
Meaning "continuous stretch" (of something) is from 1670s. That of "continuous use, circulation, or observance" (as in run of luck) is by 1714. The general sense of "a continuous series or succession" has yielded many specific meanings, as "three or more playing cards in consecutive order" (1870). In music, "a rapid succession of consecutive tones," by 1835.
The financial meaning "extraordinary series or rush of demands on a bank, etc." is recorded from 1690s. The market sense of "sustained demand for something" is by 1816.
From 1712 as "a spell of sailing between two ports;" hence also "an excursion trip" (1819); "single trip by a railroad train" (1857); the military aircraft attack sense (as in bombing run) is from 1916. Hence also "a regular round in a vehicle" (as in paper run, milk run, etc.).
In printing, the meaning "total number of copies done in a single period of press-work" is from 1909. In publishing, "set or series of consecutive numbers of a periodical," by 1889.
Meaning "tear in a knitted garment or stocking" is from 1922, probably on the notion of "a failure caused by looseness, weakness, or giving way;" to run had a specialized sense in reference to machinery, "to slip, go awry" (1846), and in reference to lace it meant "to unravel, come undone" (1878). Also compare running stitch "loose, open stitch" (1848).
Phrase a run for one's money "satisfaction for trouble taken" is from 1872 in a figurative sense, from horse racing, where it implied real competition (1841).