If you've left enough backing you certainly could turn it under and blind stitch down to the front. Or you could turn the top under a 1/4 inch and the backing under slightly behind it then blind stitch down. I've done this once on a wall hanging, must I must say I prefer the look of a separate binding; If you go that route then yes, you trim the backing and batting to match the top; Although I read on here someone who leaves a bit of batting extra - probably about 1/4 inch to pad out the binding and I plan on trying that myself next time i bind something. As for fabric - if you've got enough left then use that, otherwise buy something which tones. I seem to remember you took binding into account when buying the backing?
K x