I also do a lot of quilting after putting on the binding, although I usually finish the border quilting first. The biggest consideration is how close the quilting will be to the binding. If you are quilting very close to the binding, the thickness of the binding can get in the way of the presser foot, whether you are using a free motion foot or a walking foot. Also, if on the back your binding is stitched a little ways inside of the front binding stitching (does that make sense?), then quilting near the edge of the binding on the front will go through the binding on the back. So, if you are planning on doing very dense quilting in the border, which would probably require you to quilt close to the binding, it's probably best to remove the binding first.
BTW, I usually cut my binding on straight-of-grain and haven't had problems.
Nancy