Etymology
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loose (v.)
c. 1200, lousen, "to set free, turn loose," also "undo, untie, unfasten," from loose (adj.). Of arrows from c. 1400. Related: Loosed; loosing.
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-let 
diminutive noun-forming element, Middle English, from Old French -elet, which often is a double-diminutive. It consists of Old French diminutive -et, -ette (see -et) added to nouns in -el, which in many cases represents Latin diminutive -ellus; see -el (2)). "The formation did not become common until the 18th c." [OED].
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loose (adj.)

early 13c., lous, loos, lowse, "not securely fixed;" c. 1300, "unbound, not confined," from Old Norse lauss "loose, free, unencumbered; vacant; dissolute," cognate with Old English leas "devoid of, false, feigned, incorrect" (source of -less) from Proto-Germanic *lausaz (source also of Danish løs "loose, untied," Swedish lös "loose, movable, detached," Middle Dutch, German los "loose, free," Gothic laus "empty, vain"), from PIE root *leu- "to loosen, divide, cut apart."

Meaning "not clinging, slack" (of clothes, etc.) is from mid-15c. Meaning "not bundled" is from late 15c. Sense of "unchaste, immoral" ("lax in conduct, free from moral restraint") is recorded from late 15c. Meaning "at liberty, free from obligation" is 1550s. Sense of "rambling, disconnected" is from 1680s. As an adverb, "loosely," from 1590s. A loose end was an extremity of string, etc., left hanging; hence something unfinished, undecided, unguarded (1540s); to be at loose ends is from 1807. Phrase on the loose "free, unrestrained" is from 1749 (upon the loose). Colloquial hang loose is from 1968.

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let (n.)

"stoppage, obstruction" (obsolete unless in legal contracts), late 12c., from archaic verb letten "to hinder," from Old English lettan "hinder, delay, impede," etymologically "make late," from Proto-Germanic *latjan (source also of Old Saxon lettian "to hinder," Old Norse letja "to hold back," Old High German lezzen "to stop, check," Gothic latjan "to hinder, make late"), related to *lata-, source of late (adj.), from PIE root *‌‌lē- "to let go, slacken."

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let (v.)

Old English lætan (Northumbrian leta) "to allow; to leave behind, depart from; leave undone; bequeath," also "to rent, put to rent or hire" (class VII strong verb; past tense let, leort, past participle gelæten), from Proto-Germanic *letan (source also of Old Saxon latan, Old Frisian leta, Dutch laten, Old High German lazan, German lassen, Gothic letan "to leave, let"), from PIE *led-, extended form of root *‌‌lē- "to let go, slacken." If that derivation is correct, the etymological sense might be "let go through weariness, neglect."

The idea of slackening lies at the root of both applications of the term. When we speak of letting one go, letting him do something, we conceive him as previously restrained by a band, the loosening or slackening of which will permit the execution of the act in question. ... At other times the slackness is attributed to the agent himself, when let acquires the sense of being slack in action, delaying or omitting to do. [Hensleigh Wedgwood, "A Dictionary of English Etymology," 1859] 

He points to similar developments in French laisser "to let" from Latin laxare "to slacken," German lassen "to permit, to let," from dialectal lass "loose."

"The shortening of the root vowel ... has not been satisfactorily explained" [OED]. Of blood, from late Old English. Other Old and Middle English senses include "regard as, consider; behave toward; allow to escape; pretend;" to let (someone) know and to let fly (arrows, etc.) preserve the otherwise obsolete sense of "to cause to."

To let (someone) off "allow to go unpunished, excuse from service" is from 1814. To let on is from 1725 as "allow (something) to be known, betray one's knowledge of," 1822 as "pretend" (OED finds a similar use in the phrase never let it on him in a letter from 1637). To let out is late 12c. as "allow to depart" (transitive); intransitive use "be concluded," of schools, meetings, etc., is from 1888, considered by Century Dictionary (1895) to be "Rural, U.S." Of garments, etc., late 14c.

Let alone "abstain from interfering with" is in Old English; the phrase in the sense "not to mention, to say nothing of" is from 1812. To let (something) be "leave it alone" is from c. 1300; let it be "let it pass, leave it alone" is from early 14c. To let go is from c. 1300 as "allow to escape," 1520s as "cease to restrain," 1530s as "dismiss from one's thoughts." Let it go "let it pass, no matter" is as old as Chaucer's Wife of Bath: "But age allas Hath me biraft my beautee Lat it go, far wel, the deuel go ther with!" [c. 1395]. Let me see "show me" is from c. 1300.

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loose cannon (n.)

in the figurative sense "wildly irresponsible person, potent person or thing freed from usual restraint," by 1896; in the literal sense an object of dread on old warships; the figurative use probably arose from a celebrated scene in a popular late novel by Victor Hugo:

You can reason with a bull dog, astonish a bull, fascinate a boa, frighten a tiger, soften a lion; no resource with such a monster as a loose cannon. You cannot kill it, it is dead; and at the same time it lives. It lives with a sinister life which comes from the infinite. It is moved by the ship, which is moved by the sea, which is moved by the wind. This exterminator is a plaything. [Victor Hugo, "Ninety Three," 1874]
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let-down (n.)
also letdown, "a disappointment," 1768, from let (v.) + down (adv.). The verbal phrase is from mid-12c. in a literal sense "cause to be lowered," of drawbridges, etc.; figuratively by 1754.
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fast and loose 
described as "a cheating game played with a stick and a belt or string, so arranged that a spectator would think he could make the latter fast by placing a stick through its intricate folds, whereas the operator could detach it at once." [James O. Halliwell, "Dictionary of Archaic and Provincial Words," 1847]. The figurative sense (1550s) is recorded earlier than the literal (1570s).
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loose-leaf (adj.)
1899, of notebooks, ledgers, etc. made to allow insertion or removal of pages at will, from loose (adj.) + leaf (n.) "page of a book."
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let-up (n.)
"cessation, restraint, relaxation, intermission," 1837, from verbal phrase let up "cease, stop" (1787). In Old English the phrase meant "to put ashore" (let out meant "put to sea"). Bartlett (1848) says the noun is "an expression borrowed from pugilists."
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