I unknowingly kind of did belly wrapping after my first, having never heard of it. It felt like my insides were going to drop out of my vagina after giving birth, and it was such a disconcerting feeling. I had a couple of "bella bands" - you know, the kind you use to extend the life of your non-preggo jeans and zipper pants when you can't button them anymore? I wore both of those kind of over/under my belly, and also used the pregnancy back brace I'd been wearing in the 3rd trimester for a bit, too. I felt much better and more "secure" and like my mid section wasn't mushing all over the place.
For my 2nd birth, I did some research, and eventually ended up getting a belly band off amazon.com - can't remember if it was specifically a belly binding band, or just a post-abdominal surgery band. It worked just fine, and more convenient than layering pregnancy tummy things. I do recall that I didn't feel comfortable using it until about 3 days after the birth (too tight), and still ended up using the bella band for the first couple of days. I probably wore it for several weeks after the birth. I felt like it was worth the money for me.
I got a moby wrap to carry my second kiddo in, and though it seems pretty large, I wonder if that might work for the first few days. I will probably just be wearing my pregnancy dresses for the first week or so anyway since they are the most comfortable thing for me right now, so I don't care too much about the bulk.
As for the question on shrinking hip width, I have no idea. I didn't use belly binding for that purpose, and guess I just don't care enough anyway!