The nocturnal flying mammal was originally not a bat but a ‘back’. The earliest form, adopted in the early Middle Ages from a Scandinavian word, was altered to bat in the 16th century, perhaps influenced by Latin batta or blacta ‘insect that shuns the light’. The creature has inspired numerous expressions. You could be as blind as a bat from the 16th century—before then the standard comparison was with a beetle. From the early 20th century you could have bats in the belfry, ‘be mad’, or, in the same vein, be bats or batty. The first recorded example of like a bat out of hell, ‘very fast and wildly’, is from the Atlanta Constitution of 3 February 1914: ‘One day we saw an automobile go down the street like a bat out of hell and a few moments later we heard that it hit the last car of a freight train at the grade crossing.’ An old-fashioned name for a bat is flittermouse [M16th], meaning literally ‘flying mouse’. Dutch vledermuis and German Fledermaus are matching terms in other languages.
The other bat, for hitting a ball, is a word adopted from French in the Old English period, and is related to battery. If you do something off your own bat you are using a cricketing phrase; it originally referred to the score made by a player's own hits, and so ‘at your own instigation’. But if you did something right off the bat, ‘at the very beginning, straight away’, you would be taking a term from baseball.
Batman has been a comic character and superhero since 1939. The less glamorous batman [M18th] is a British army officer's personal servant. This bat is unrelated to the other two. It came through French from medieval Latin bastum ‘a packsaddle’ (see bastard) and originally referred to a man in charge of a bat-horse, which carried the luggage of military officers.