bat (n.1)

"a stick or staff used in beating, a war-club, staff used to strike the ball in certain games," c. 1200, from rare Old English batt "cudgel," a western England word at first, probably from Welsh or another Celtic source (compare Irish and Gaelic bat, bata "staff, cudgel"), later reinforced and influenced by Old French batte "pestle," from Late Latin battre "to beat;" all from PIE root *bhat- "to strike." As a kind of wooden paddle used to play cricket (later baseball), it is attested from 1706.

A Middle English sense of "a lump, piece, chunk" (mid-14c.) was used in reference to bread, clay, wool, and survives in brickbat and batting (n.1). 

The phrase right off the bat (1866), also hot from the bat (1870), probably represent a baseball metaphor, but cricket or some other use of a bat might as easily be the source—there is an early citation from Australia (in an article about slang): "Well, it is a vice you'd better get rid of then. Refined conversation is a mark of culture. Let me hear that kid use slang again, and I'll give it to him right off the bat. I'll wipe up the floor with him. I'll ---" ["The Australian Journal," November 1888].

flying mouse-like mammal (order Chiroptera), 1570s, a dialectal alteration of Middle English bakke (early 14c.), which is probably related to Old Swedish natbakka, Old Danish nathbakkæ "night bat," and Old Norse leðrblaka "bat," literally "leather flapper," from Proto-Germanic *blak-, from PIE root *bhlag- "to strike" (see flagellum).

If so, the original sense of the animal name likely was "flapper." The shift from -k- to -t- may have come through confusion of bakke with Latin blatta "moth, nocturnal insect."

The Old English word for the animal was hreremus, from hreran "to shake" (see rare (adj.2)), and rattle-mouse, an old dialectal word for "bat," is attested from late 16c. Flitter-mouse (1540s) occasionally is used in English (with variants flinder-mouse, flicker-mouse) in imitation of German fledermaus "bat," from Old High German fledaron "to flutter."

As a contemptuous term for an old woman, it is perhaps a suggestion of witchcraft (compare fly-by-night), or from bat as "prostitute who plies her trade by night" [Farmer, who calls it "old slang" and finds French equivalent "night swallow" (hirondelle de nuit) "more poetic"].

"to move the eyelids," 1847, American English, an extended sense from the earlier meaning "flutter (the wings) as a hawk" (1610s), a variant of bate (v.2). Related: Batted; batting.

"to hit, beat, or strike with a bat," mid-15c., from bat (n.1). Specifically as "to strike a ball with a bat" from 1745. Related: Batted; batting.