"edged instrument for hewing timber and chopping wood," also a battle weapon, Old English ces (Northumbrian acas) "axe, pickaxe, hatchet," later x, from Proto-Germanic *akusjo (source also of Old Saxon accus, Old Norse ex, Old Frisian axe, German Axt, Gothic aqizi), from PIE *agw(e)si- "axe" (source also of Greek axine, Latin ascia).
The spelling ax is better on every ground, of etymology, phonology, and analogy, than axe, which became prevalent during the 19th century; but it is now disused in Britain. [OED]
The spelling ax, though "better on every ground, of etymology, phonology, & analogy" (OED), is so strange to 20th-c. eyes that it suggests pedantry & is unlikely to be restored. [Fowler]
Meaning "musical instrument" is 1955, originally jazz slang for the saxophone; rock slang for "guitar" dates to 1967. To have an axe to grind is from a Sept. 7, 1810, essay in the Luzerne (Pennsylvania) "Gleaner" by U.S. editor and politician Charles Miner (1780-1865) in which a man flatters a boy and gets him to do the chore of axe-grinding for him, then leaves without offering thanks or recompense. It was published in a collection in 1815 titled "Essays From the Desk of Poor Robert the Scribe." The story ("Who'll Turn the Grindstone?") has been misattributed since late 19c. to Benjamin Franklin, a mistake continued in Weekley, OED print edition, "Century Dictionary," and many other sources (Bartlett's "Familiar Quotations" has gotten it right since 1870).
Old English giefan (West Saxon) "to give, bestow, deliver to another; allot, grant; commit, devote, entrust," class V strong verb (past tense geaf, past participle giefen), from Proto-Germanic *geban (source also of Old Frisian jeva, Middle Dutch gheven, Dutch geven, Old High German geban, German geben, Gothic giban), from PIE root *ghabh- "to give or receive." It became yiven in Middle English, but changed to guttural "g" by influence of Old Norse gefa "to give," Old Danish givæ.
Meaning "to yield to pressure" is from 1570s. Give in "yield" is from 1610s; give out is mid-14c. as "publish, announce;" meaning "run out, break down" is from 1520s. Give up "surrender, resign, quit" is mid-12c. To give (someone) a cold seems to reflect the old belief that one could be cured of disease by deliberately infecting others. What gives? "what is happening?" is attested from 1940. To not give a (some thing regarded as trivial and valueless) is from c. 1300 (early examples were a straw, a grass, a mite).
"ice-axe used by Alpine climbers," 1868, from Savoy French piolet "climber's ice-axe" (19c.), diminutive of piolo "axe," which is perhaps from Medieval Latin piola "plane, scraper."