I have done many quilts with foldover binding, mainly charity ones, since it is quicker for me.
I cannot tell from your picture if you have a border. If not, you will have the problem of the quilting stitches going all the way to the edge and need to be secured. If you have a border with no quilting running to edge, there is no problem.
I understand about accidently cutting the backing and have done it a few times. It is important to fold the backing completely out of the way while trimming the quilt and then bringing the backing back in place to trim it. I use the rotary cutter and ruler. Using a scissors wouldn't work for me.