bat (n.1) Look up bat at Dictionary.com
"a stick, a club," Old English *batt "cudgel," perhaps from Celtic (compare Irish and Gaelic bat, bata "staff, cudgel"), influenced by Old French batte, from Late Latin battre "beat;" all from PIE root *bhat- "to strike." Also "a lump, piece" (mid-14c.), as in brickbat. As a kind of paddle used to play cricket, it is attested from 1706.

Phrase right off the bat is 1888, also hot from the bat (1888), probably a baseball metaphor, but cricket is possible as a 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].
bat (n.2) Look up bat at Dictionary.com
flying 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 "leather flapper" (for connections outside Germanic, 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."

Old English word for the animal was hreremus, from hreran "to shake" (see rare (adj.2)), and rattle-mouse is attested from late 16c., an old dialectal word for "bat." As a contemptuous term for an old woman, 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"].
bat (v.1) Look up bat at Dictionary.com
"to move the eyelids," 1847, American English, from earlier sense of "flutter as a hawk" (1610s), a variant of bate (v.2) on the notion of fluttering wings. Related: Batted; batting.
bat (v.2) Look up bat at Dictionary.com
"to hit with a bat," mid-15c., from bat (n.1). Related: Batted; batting.