I have made several quilts with Minkee. That Minkee is stronger than any binding you could put on there. Just bring the Minkee up over the quilt edges to the top. I usually have about a one inch hem and it's easier for me to just use a straight stitch to sew it down. I have seen other use decorative stitches. Your quilt, your call. The one I made for my youngest over 5 years ago shows no wear and tear on the edge of the quilt and he sleeps all twisted up in it every night.
We used crochet thread to tie quilts when I was a kid. Comes in several colors. If you are using Minkee, you are probably not using batting - would make it too heavy and hot. We did ties at every 4 inches. But, did you know you could just use a decorative stitch on your machine to tie/tack your quilt? Find a pretty one like a flower or some other design and put it about every 4 to 6 inches.
I have heard others using yarn to tie with.