R.,
I know how worried you can be about your 3 year old daughter and her sling. The position of her arm while she sleeps would depend how old the fracture is. If it is a fairly new fracture, less than 2 weeks old - positioning is very important. But the sling is provided to help your daughter hold her arm up while she is sitting or standing. This will help with providing the proper angle for the elbow to heal properly. If you leave it without the sling - during her awake time -it may get to heavy for one and two, her muscles and tendons around the area may become sore too -they are also healing.
As for when she sleeps, you can take the sling off - and prop her arm up on a small pillow. Align her wrist/elbow area (straight) - over her belly or by her side. This will allow for proper circulation. Yes, she will move, jiggle all around the bed. But, don't worry - just reposition the arm. She is only 3 years old. I don't think the doctor will expect anything more. If you do notice swelling to her fingers, lover her elbow more than her hand, this usually helps. Call her doctor if she has TOO much swelling or discomfort though. And remember, try not to let her put anything inside her cast - that will just leads to a whole lot of other issues . Don't worry - you both will do fine.
Good luck....