My 14-year-old daughter is mean to her 3-year-old sister. She shouts at her and gets mad at her for wanting to watch the same video over and over in her room. She doesn't like to share with her, but wants her little sister to share her little things with her. They share the same room for now until we move. My older girl hates to take her sister over to her friends' house so when she goes. I say to her: "If she can't go, you can't go." I see her friends with their younger siblings. Am I wrong for making her do this?
I know you want your daughters to care for each other, but by forbidding your older girl to visit her friends unless she takes her sibling is setting her up to resent the little sister you want her to care for. Your oldest daughter can't express her anger towards you, so of course she is going to take it out on her younger sister, who she sees as the cause of her pain.
What does a 3-year-old have in common with a 14-year-old? Would you want to watch the same video over and over again? Do you want your 3-year-old involved in teenage activities? Your insistence on forcing them together is harmful for both girls. My experience tells me that if you let the girls naturally share family activities -- meals, vacation time -- the "meanness" will disappear.
Because of the age differences it will be very difficult, if not impossible, for the girls to relate to each other in the close relationship. But they can develop a truly loving sisterly relationship if you let them develop it naturally and understand that each is at a different developmental stage. You say they won't have to share a room after you move. Stick to that promise and if at all possible separate them now. Your teen doesn't have enough space if she is sharing a room with a three-year-old.