shoot-out n.

Middle English sheten "hasten from place to place; move swiftly; thrust forward; discharge a missile, send an arrow from a bow," from Old English sceotan (class II strong verb; past tense sceat, past participle scoten), "dart forth, go swiftly and suddenly," also "discharge (a missile or weapon);" also, of a person, "go suddenly from place to place;" also transitive "send out or forth with sudden or violent motion; put forth or extend in any direction; strike with anything shot."

This is from Proto-Germanic *skeutanan (source also of Old Saxon skiotan, Old Norse skjota "to shoot with (a weapon); shoot, launch, push, shove quickly," Old Frisian skiata, Middle Dutch skieten, Dutch schieten, Old High German skiozan, German schießen), often said to be from PIE root *skeud- "to shoot, chase, throw," but Boutkan gives it no IE etymology.

The sense of "dart along" (as pain through the nerves or a meteor in the sky) is by late 13c.; that of "come forth" (as a plant) is by late 15c. As "increase rapidly, grow quickly" by 1530s (often with up (adv.)). By 1690s as "be emitted in rays or flashes" (as light is); by 1530s in weaving, "variegate by interspersing colors."

The general sports sense of "kick, hit, throw etc. toward the goal" is by 1874. In reference to pool playing, by 1926. The meaning "strive (for)" is by 1967, American English. The sense of "descend (a river) quickly" is from 1610s. The slang meaning "to inject by means of a hypodermic needle" is attested by 1914 among addicts. The meaning "to photograph" (especially a movie) is from 1890.

As an interjection, an arbitrary euphemistic alteration of shit, it is recorded by 1934.

Shoot the breeze "chat" is attested by 1938 (as shooting the breeze), perhaps originally U.S. military slang. Shoot to kill is attested from 1867. Slang shoot the cat "vomit" is from 1785.

To shoot the moon formerly meant "depart by night with ones goods to escape back rent" (c. 1823).

O, 'tis cash makes such crowds to the gin shops roam,

And 'tis cash often causes a rumpus at home ;

'Tis when short of cash people oft shoot the moon ;

And 'tis cash always keeps our pipes in tune.

Cash! cash! &c.

["The Melodist and Mirthful Olio, An Elegant Collection of the Most Popular Songs," vol. IV, London, 1829]

Shoot against the moon was used by Massinger  (1634) as a figure of an impossible attempt.

expressing motion or direction from within or from a central point, also removal from proper place or position, Old English ut "out, without, outside," from Proto-Germanic *ūt- (Old Norse, Old Frisian, Old Saxon, Gothic ut, Middle Dutch uut, Dutch uit, Old High German uz, German aus), from PIE root *uidh- "up, out, up away, on high" (source also of Sanskrit ut "up, out," uttarah "higher, upper, later, northern;" Avestan uz- "up, out," Old Irish ud- "out," Latin usque "all the way to, continuously, without interruption," Greek hysteros "the latter," Russian vy- "out").

Sense of "to a full end, completely, to a conclusion or finish" is from c. 1300. Meaning "so as to be no longer burning or alight; into darkness" is from c. 1400. Of position or situation, "beyond the bounds of, not within," early 15c. Meaning "into public notice" is from 1540s; that of "away from one's place of residence," c. 1600. The political sense of "not in office, removed or ejected from a position" is from c. 1600. Meaning "come into sight, become visible" (of stars, etc.) is by 1610s. In radio communication, a word indicating that the speaker has finished speaking, by 1950.

As a preposition, "out of; from, away from; outside of, beyond; except; without, lacking;" mid-13c., from the adverb.

Meaning "from harmonious relations, into quarreling" (as in to fall out) is from 1520s. Meaning "from one's normal state of mind" (as in put out) is from 1580s; out to lunch "insane" is student slang from 1955. Adjectival phrase out-of-the-way "remote, secluded" is attested from late 15c. Out-of-towner "one not from a certain place" is from 1911. Out of this world "excellent" is from 1938; out of sight "excellent, superior" is from 1891. To (verb) it out "bring to a finish" is from 1580s. Expression from here on out "henceforward" is by 1942. Out upon, expressing abhorrence or reproach, is from early 15c.