Yes, you should paste up or glue the spine leather directly to the spine to match historical examples. Hollow backs are a fairly recent thing in the bookbinding timeline.
The leather doesn't need much dampness if you want to go that route. I would soak a sponge, squeeze out most of the water, then give the skin a wash, then immediately paste it up for covering on the other side and go at it. Be careful not to stretch the leather too much over the boards, or they'll warp severely when it dries, and it'll be a pain to get them to uncurl again. If that happens, though, and the boards are being uncooperative after pressing, use a hair dryer on low and wave it slowly across the inside of the boards about 2 inches away from the surface. They'll begin to uncurl again.
The dampness will aid you in forming the bands on the spine. If you don't have band nippers you can shape them out with the edge of a bonefolder, but you'll have to spend a little time with it -- which is fine. Don't rush through the covering process. Take your time and make sure it looks good to you.
If you're unable to make a rounded spine, you can get a decent round just by applying the right tension to the cords when laced into the cover boards. If you think "tight but not too tight," that's the right idea, and you'll probably get a nice subtle curve. As for the hinge area, I would bevel the bottom, innermost edge of the board with a wood rasp. Generally if you're making a rounded spine you'll have shoulders for those boards, but doing it this way with the edge beveled will help the board hug the textblock and relieve some strain when opening the book.
Having seen the Lost Gutenbergs in person, I can tell you that those books are built like battleships, and their construction is a bit too complicated to explain here. It's something that needs to be shown. I personally am still learning all the details. I can tell you, though, that the first thing done in covering them is the spine. I do this sometimes with large books -- glue up the spine leaving small areas at the head and tail unglued for the turn-ins later, get the spine adhered and bands formed, then once well attached paste up the rest and finish covering.