1. I save decent sized pieces of batting to use for smaller projects like hot pads, pot holders and table runners. I save some smaller size ones to fill handmade pin cushions with...I call them pin pillows.
2. I try to pull my backs as tight as possible without stretching them. I then smooth out as I go, keeping it taut, otherwise it will pucker. I think you are making your backings fine.
3. As far as the doll quilts go, I would use your little tiny leftover pieces of batting and make small stuffed animals to go with your doll quilts. My son was in Children's hospital for quite some time and he got a little stuffed bear that he loves dearly. Being a mom with a child in the hospital, it was so nice to think of a stranger thinking of my sick baby. A little something for him was so special. The auxillary club provided little things for siblings too. Someone had embroidered "I'm with the patient" on the chest of many of these animals so their siblings could get something too.....just a thought.