"give a small present of money to," c. 1600, originally "to give, hand, pass," thieves' cant, perhaps from tip (v.3) "to tap." The meaning "give a gratuity to" is first attested 1706. The noun in this sense is from 1755; the noun meaning "piece of confidential information" is from 1845; and the verb in the sense "give private information to" is from 1883.
The popularity of the tale of the word's supposed origin as an acronym in mid-18th century English taverns seems to be no older than Frederick W. Hackwood's 1909 book "Inns, Ales and Drinking Customs of Old England," where it was said to stand for To insure promptitude (in the form to insure promptness the anecdote is told from 1946). A reviewer of the book in The Athenaeum of Oct. 2, 1909, wrote, "We deprecate the careless repetition of popular etymologies such as the notion that "tip" originated from an abbreviated inscription on a box placed on the sideboard in old coaching-inns, the full meaning of which was "To Insure Promptitude." Also see here.
by c. 1200 as an emphatic form of Old English of (see of), employed in the adverbial use of that word. The prepositional meaning "away from" and the adjectival sense of "farther" were not firmly fixed in this variant until 17c., but once they were they left the original of with the transferred and weakened senses of the word. Meaning "not working" is from 1861.
Off the cuff "extemporaneously, without preparation" (1938) is from the notion of speaking from notes written in haste on one's shirt cuffs. In reference to clothing, off the rack (adj.) "not tailored, not made to individual requirements, ready-made" is by 1963, on the notion of buying it from the rack of a clothing store; off the record "not to be publicly disclosed" is from 1933; off the wall "crazy" is 1968, probably from the notion of a lunatic "bouncing off the walls" or else in reference to carom shots in squash, handball, etc.