Baseball and cricket both have origins in the same primitive games (see link). In addition, several early American baseball pioneers (notably George Wright) had a history with 19th century cricket and encouraged the continued use of the terms.

Maybe because the game of Cricket has been around since at least the late 1400’s?

Because a bat is used to Bat or strick the ball. Ever see a cat bat a mouse or toy? Bat comes from Old English “batt”

England created the sport and America just made it better.

1. Baseball developed from cricket.
2. This is the U.S., we can call it anything we darn well please.