shotgun n.

Middle English shot "a missile, arrow, dart" (senses now archaic or obsolete); "a swift movement, a gushing out," from Old English scot, sceot "a shot, a shooting, an act of shooting; that which is discharged in shooting, what is shot forth; darting, rapid motion."

This is from Proto-Germanic *skutan (source also of Old Norse skutr, Old Frisian skete, Middle Dutch scote, German Schuß "a shot"), from PIE root *skeud- "to shoot, chase, throw." The Old English noun is related to sceotan "to shoot." The meaning "discharge of a bow, missile," also is from related Old English gesceot.

The noun was extended to other projectiles (balls, bullets) by mid-15c. Especially "lead in small pellets, a small ball or pellet," a number of which are combined in one charge, which is attested by 1770 (shortened from earlier small shot, 1727).

The general sense of "an attempt to hit with a projectile" is by 1650s. Extended to sports (hockey, basketball, etc.) by 1772, originally in curling. It is attested by early 15c. as "range or distance of a missile in flight," hence "range" in general (c. 1600), as in earshot. 

Another original meaning, "payment" (perhaps literally "money thrown down") is preserved in scot-free; also see scot (n.). The notion of "throwing down" might have led to the meaning "a drink," first attested 1670s; the more precise meaning "small drink of straight liquor" is by 1928.

The sense of "hypodermic injection" is attested from 1904; the figurative phrase shot in the arm "stimulant" is by 1922. The broad meaning "a try, an attempt" is by 1756; the sense of "remark meant to wound" is by 1841. The meaning "an expert in shooting with a firearm" is from 1780; the sense of "a rocket flight" is by 1934. The camera-view sense is by 1958.

To call the shots "control events, make decisions" is American English, 1922, perhaps from sport shooting. Shot in the dark "uninformed guess, random attempt" is by 1885. Big shot "important person" is from 1861. 

By the rude bridge that arched the flood,

   Their flag to April’s breeze unfurled,

Here once the embattled farmers stood

   And fired the shot heard round the world.

[Emerson, from "Concord Hymn"]  

mid-14c., gunne "an engine of war that throws rocks, arrows or other missiles from a tube by the force of explosive powder or other substance," apparently a shortening of woman's name Gunilda, found in Middle English gonnilde "cannon" and in an Anglo-Latin reference to a specific gun from a 1330 munitions inventory of Windsor Castle ("... una magna balista de cornu quae Domina Gunilda ..."). Also compare gonnilde gnoste "spark or flame used to fire a cannon" (early 14c.).

The woman's name is from Old Norse Gunnhildr, a compound of gunnr and hildr, both meaning "war, battle." First element from PIE *gwhen- "to strike, kill" (see bane); for second, see Hilda.

The identification of women with powerful weapons is common historically (such as Big Bertha, Brown Bess, Mons Meg, etc.).

Or perhaps gun is directly from Old Norse gunnr "battle." The word was perhaps influenced by or confirmed by (or possibly from) Old French engon, dialectal variant of engin "engine."

Meaning grew with technology, from cannons to firearms as they developed 15c.; popularly applied to pistols and revolvers from 1744. In modern military use the word is restricted to cannons (which must be mounted), especially long ones used for high velocity and long trajectory. Hence great guns (1884 as an exclamation) distinguished from small guns (such as muskets) from c. 1400. Meaning "thief, rascal" is from 1858. For son of a gun, see son. To jump the gun (1912, American English) is a figurative use from track and field. Guns "a woman's breasts" (especially if prominent) attested by 2006.

[G]un covers firearms from the heaviest naval or siege guns (but in technical use excluding mortars and howitzers) to the soldier's rifle or the sportsman's shotgun, and in current U.S. use even the gangster's revolver. In the other European languages there is no such comprehensive word, but different terms for the small or hand gun of the soldier or sportsman (even these, sometimes differentiated) and the heavy naval guns or artillery pieces .... [Buck, 1949]

"portable firearm having a barrel or barrels with a spirally grooved bore," by 1775; the word was used earlier of the grooves themselves (1751), and is a noun use from rifled pistol, 1680s, from the verb rifle meaning "to cut spiral grooves in" (a gun barrel); see rifle (v.2).

The spirals impart rotation to the projectile, making its flight more accurate. Rifles "troops armed with rifles," sometimes as part of a unit name, is by 1843. Rifle-range is from 1850 as "distance a rifle-ball will carry" (also, and earlier rifle-shot, 1803); the meaning "place for rifle shooting" is by 1862. Rifle-ball is by 1884; the  word continued in use after cylindrical bullets with conical heads replaced round ones.