My guess is Saddlebred/Quarter Horse mix.
About the sway back, I bought a horse a few months ago who looked pretty sway backed. He started looking better as he gained weight, but he is really improving now that I've been working to strengthen those back muscles with dressage "collecting" exercises. I think he just had not been ridden much for a few years and his back muscles had gotten very weak.
Start slow with him. Be sure he's being fed well and is otherwise healthy, then research exercises you can do with him to gradually strengthen his back. His back is very long, which doesn't help but who knows?
Can he lift his back up if he is motivated? My horse will lift his back up almost to a level position when I squirt his tummy with the hose when I'm washing him. If he CAN lift his back, then there is a lot of hope that you can strengthen the back muscles and minimize the sway back. I would probably not put a heavy rider on him or otherwise stress his back muscles, especially before you work to strengthen them.