"edged instrument for hewing timber and chopping wood," also a battle weapon, Old English æces (Northumbrian acas) "axe, pickaxe, hatchet," later æx, 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]
The 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).