The creator of Catwoman says that he picked a cat as he felt that they were an antithesis of bats.
"I felt that women were feline creatures and men were more like dogs. While dogs are faithful and friendly, cats are cool, detached, and unreliable. I felt much warmer with dogs around me—cats are as hard to understand as women are. Men feel more sure of themselves with a male friend than a woman. You always need to keep women at arm's length. We don't want anyone taking over our souls, and women have a habit of doing that. So there's a love-resentment thing with women. I guess women will feel that I'm being chauvinistic to speak this way, but I do feel that I've had better relationships with male friends than women. With women, once the romance is over, somehow they never remain my friends"
Batman #62 (December 1950) reveals that Catwoman was an amnesiac flight attendant who turned to crime after suffering a prior blow to the head during a plane crash she survived. She reveals this in the Batcave after being hit on the head by a piece of rubble while saving Batman while he was chasing her. However, in issue 197 of The Brave and the Bold, she later admits that she made up the amnesia story because she wanted a way out of the past life of crime.
She uses a whip because it is a weapon that someone must be trained to use. She also uses a pistol if she doesn't have her whip.
As for her costume, she didn't have it in Batman#1, #2 and #3. In #3, she just has a fur mask.