Wendy, when I wet my quilts it is usually in the washing machine. I use the spin cycle to get a lot of the water out. sometimes I dry the quilt in the dryer so it's nearly dry. sometimes I lay out the quilt on a plastic tarpaulin and thick towels on the living room floor, patting it into place.
The wrinkles that appear in the quilt are from shrinkage, which I actually like the look of. If you want a crisp, unwrinkled surface on your quilt probably the best thing to do is keep it away from water.
I've never ironed a quilted quilt, so don't have any info on how that would turn out.I don't have a carpet to pin a quilt into, can't make any comments on how that goes either
Kathy