I have seen some lap quilts where the bottom 2 corners are cut at a diagonal about 6" from the corner...this keeps the corners from getting caught in the wheels and doesn't affect the functionality of the quilt...flannel backing sounds like a great idea.
What a sweet idea for your aunt...it might be nice to add a variety of textures...satin, fleece, etc near the area where her hands would lay, just to give her the ability to feel something soft and different.