I think many people get a bit overexcited about children lying. In this case, from the behaviour you are describing, your dd is not attempting to deceive you. When she denies doing something which you saw her do, and which she must KNOW you saw her do, she cannot be expecting you to believe her. She's hardly on the road to delinquency.
Her protests are about something else. Maybe she is pushing your buttons because she wants an argument, maybe she is trying to make everything better by pretending to herself that it didn't happen. I don't know. Perhaps you can figure out what is behind her fibs.
But I think you are getting sidetracked from the behaviour you really want to change. Perhaps instead of accusing her of doing something wrong (which seems to invite her denials and your consequent anger), you could just calmly tell her what you want her to do?
For instance, when she bangs on the bed: "Can you please move away from the bed." and when she protests, "But I wasn't DOING anything?" you can reply, "I didn't say you did anything. I asked you to move away from the bed. Can you go brush your teeth please?"
or if she hurts her dad: "Go and sit on the sofa for a bit. Daddy is getting hurt." "I DIDN'T HURT HIM! WE WERE JUST PLAYING. He started it." "Well, he will be happier if you sit on the sofa for a while. He is getting cross. Please go to the sofa now."