My bit of experience tells me that you might have that stiffness for all the reasons you stated but it shouldn't be a problem if you wash the quilt when you are done binding it.
As far as the quilt going off square. This will happen and I never square up a quilt until after it is quilted. If you like to baste the edge before you start quilting, I would remove it in sections as I worked to the edge of the quilt just like removing basting stitches so the fabric can move out from the center. Even with basting the batting will want to shift a bit as it is worked.
one of my tricks is to make any borders an inch or two wider than What I want it to finish at and then I have enough to trim and square up at the of quilting.