Learn English – What are the origins for the phrases “Knock it off” and “Cut it out”

Bartlett's Dictionary of Americanisms, 1848, s.v. Drag out says ‘A “knock down and drag out” is a fight carried to extremities [my emphasis]. The term drag out seems also to be used, at the South, to denote a bully, a tearer’, and gives an instance:

Set to your partner, Dolly,—Cut him out, Jim, —Sald does put her foot down good. The yellow roan’s up! He a rael [sic] stormer, ring clipper, snow belcher, and drag out —Southern Sketches

I have always taken the phrase in more or less Chris’ sense: that a drag-out fight is one in which the defeated party is not only unable to rise for thirty seconds but remains indefinitely unconscious, so that he must be dragged out of the ring.

Early uses, however, show that drag out is an action performed by a fighter on his opponent:

A few days since, says the Rochester Republican, a clever good-for-nothing fellow came to our village, from the country, and being somewhat taken with its appearance, and feeling “pretty cute,” took too much of “the creature;” the consequence of which was, a little zig-zag in his movements, and a design to “knock down and drag out.” —Ladies Museum, Providence [, R.I.], Jan 21, 1826

... one of those dashing blades, so frequently to be encountered in the southern country, who, despising the humdrum monotony of regular life, are ready for adventure—lads of the turf—the muster-ground—the general affray—the men who can whip their weight in wild cats—whose general rule it is to knock down and drag out. —W. Gilmore Simms, Guy Rivers: A Tale of Georgia, 1834

“Once, my son, old Jim could knock down, drag out, whip, lift, or throw any man in all Sangamon” —Edmund Flagg, The far West, 1838

I can knock down and drag out a whole regiment, whip my weight in Indians, swallow a buffalo whole, and pick my teeth with the bones. —“A Duel in Georgia”, The London Journal, No.172, Vol.VII, June 10, 1848

The grisly character of that performance—and the ‘extremities’ to which a fight might be carried—is suggested in Richard Penn Smith's Colonel Crockett's Exploits and Adventures in Texas ... Written by Himself (1837). On the road to Texas Crockett hears a fight in the distance, and arrives at the last moment:

I saw the uppermost one (for I could not see the other) make a heavy plunge with both his thumbs, and at the same instant I heard a cry in the accent of keenest torture, “Enough! My eye is out.”  [...]
 ”Now blast your corn-shucking soul,” said the victor, a lad about eighteen, as he rose from the ground, “come cutt’n your shines ‘bout me agin, next time I come to the Court House, will you! —Get our owl-eye in agin if you can.

It turns out that the ‘fight’ was no such thing, but a solo affair in which “the young man had played all the parts for his own amusement”; and the author says that

All that I had heard and seen was nothing more or less than what is called a rehearsal of a knock-down and drag-out fight.

Unlikely as it may seem, the notion that drag-out originally signified mutilation appears to be corroborated in a US House of Representatives speech defying certain proposals by members from Kentucky:

These men never had a meeting some few years ago, as travellers in these two districts relate, either for sacred or secular purposes, but that, when the service was over, they always took a set to, a knock down and drag out. It has been said to be a fact, that twenty years ago, in either of those districts, whenever such a festival had happened over night, boys might, in the morning, gather up several baskets full of eyes, noses, ears, and other members subjected to depredation by the established laws of combat. —Speech of Mr. Burges, of Rhode Island..., 1828.

I find it interesting that the two earliest instances of drag-out I have found, from 1826 and 1828, both originate with Rhode Island sources.

"To come off" has multiple meanings, most dictionaries list more than ten different meanings depending on the area in which it is used. When speaking of theatre you can say "The play came (well) off, meaning it was a success. Sometimes it is a bit difficult to explain a semantic development, then it is only possible to use associations. Well, to come off in this usage reminds me a bit of shooting. A shot or the bullet can come well off/out of the gun or sometimes the bullet can explode in the barrel.

The other development of meaning, when speaking of a sports event, is in principle similar. "The match came off on Tuesday", meaning took place, can be understood when you think of "got started and finished". There was no hindering cause that prevented the match. Well, that's my best try to explain such tricky semantic shifts at short notice. Perhaps someone can do it better than I can.