Maybe she honestly does not see her sister as really being human with feelings and alive. She sees her more like a plaything that does not really get hurt.
3 years old is still young. In preschool they push and shove get tired of toys and throw then push them away etc.. They are learning that each of them have feelings. 3 year old are very self centered. They care about themselves, want instant gratification and needs to be taught empathy..
What are you doing about discipline besides talking? I would think taking the book away until she could be sorry to her little sister or until sister was finished with the books.
Pushing sister down the stairs is an "I am sorry sister" and a time out, siting quietly to help her calm down and think about "what it would feel like if someone pushed her down the stairs?" When time out is over she should apologize to sister and promise not to do that anymore.
Pushing sister off of the couch, means going out of the room and sitting in the hall by herself till little sister is calm, comforted and safe.
I would think it also makes mom mad that her behavior seem so mean to sister. And you know she can do better.
Make her more responsible for helping with baby sister. Going up the stairs? Ask her to hold little sisters hand while they walk up together. Have her "read a book" to little sister, while little sister looks at her own book.
Explain what little sister can do or what she cannot do, so your daughter understands.. And then also let her have some time at home without baby sister always having to be around her.
Let her help you or dad do the grown up things you all do.. Let her have your attentions without baby sister.
Praise her when she is good with baby sister. I like how you are sharing your snack with your sister. Thanks you for holding baby sisters hand. Thank you for helping her go up the stairs. Thank you for carrying her diaper bag.
Let her know when baby sister is watching and learning from her. Look baby sister likes how you can dress yourself. Maybe when she is 3 you can teach her how to put on her pants.
In her mind babies are tiring, they take up so much attention. we cannot go and blow as easily because of the baby.. everybody seems so concerned about the baby, but sometimes, she just gets tired of it. She is 3, it will get better. I promise, the first time she sees someone else being mean to her sister, she will be all over them.
It is fairly normal.