Yes I have used minky as backing but when I do I nix the batting. It's just too thick and heavy to quilt through.
Yes, I wash the minky also and dry it ahead of time, and all my cotton fabric is usually pre-washed before I start making the cotton pieced top.
Talk to your long armer before using it if you aren't planning on quilting it yourself so you don't end up in a pickle trying to get it quilted.
Hope that helps!