I have only used Minky on a long arm, and I had very few problems. Minky stretches, but with all fabrics that stretch, one way is worse than the other, so I set it up across grain on the long arm. I took over the project from a friend of mine, she had started one of two (for twins!) quilts in the other orientation and left it on the machine for a long time and the fabric had stretched badly. I had to take out what my friend had done, but again, once I turned the fabric it was better and still usable.
I have another friend who loves Minky but her quilting machine does not have a deep enough throat to handle more than a baby sized project. Her machine quilter has no problems with Minky and no extra charges for using it.
With both Minky and fleece backings, having tried without batting I prefer a thin batt, or at least a piece of flannel in between. Something for the quilting stitches to bite into.