"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).
also pick-axe, "tool used for breaking up and digging ground," especially a pick with a sharp point on one side of the head and a broad blade on the other, early 15c., folk etymology alteration (by influence of axe) of Middle English picas, pikeis (mid-13c.), via Anglo-French piceis, Old French pocois (11c.) and directly from Medieval Latin picosa "pick," which is related to Latin picus "woodpecker" (see pie (n.2)).
"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."
medieval weapon (a broad blade with sharp edges, ending in a point and mounted on a long handle), late 15c., from French hallebarde (earlier alabarde, 15c.), from Middle High German halmbarte "broad-axe with handle," from halm "handle" (see helm) + barte "hatchet," from Proto-Germanic *bardoz "beard" (see beard (n.)), also "hatchet, broadax" ("because the actual axe looks like a beard stuck to the wooden handle" - Boutkan). An alternative etymology [Kluge, Darmesteter] traces first element to helm "helmet," making the weapon an axe for smashing helmets. In 15c.-16c. especially the arm of foot-soldiers.