I never iron my binding in half. I sew the binding on with a machine on the front and then roll it to the back and hand stitch. I find it much easier to hand stitch if it hasn't been ironed with a sharp crease. My stitches do not show and with a sharp crease it is much harder to run your needle through without the stitch showing.
I usually don't have patchwork next to my binding (usually a border) so I don't have to be true to 1/4" seam allowance. My the edge of my walking foot is 3/8" so that's what I use with a 2 1/2" binding fabric. If I use 1/4" seam allowance then I use 2 1/4" binding fabric. To miter the corner I just stop 3/8" instead of 1/4" but otherwise proceed as usual.
Before I put on the binding I stitch very close to the edge all the way around my quilt and then trim the excess batting and backing away. I also square my quilt at this point although that is usually only minimal adjustment because of the way I have measured and put on my borders. The stitching gives the edge some stability and it's one less thing I have to worry about when putting on the binding that all the layers are even.
I piece my bindings using a bias seam. This reduces bulk and makes the seam almost invisible. I audition the binding roughly before sewing it on to make sure seams do not come in corners.
I have never had much luck making bindings that are machine sewn on both sides look very neat. I like hand sewing so I don't mind that part of it.