I work in alligator a lot, however, I'm a watch strap maker, so not all of this may be relevant.
Cutting-rotary, the only way to go
Watch out for excess, loose flesh on the backside of the hide. This will cause trouble later as it leads to separation of glued pieces. Shave a little bit off with a splitter if you have one, otherwise a sanding block works for larger pieces, and you can use a dremel with sanding drum for small pieces.
If you having problems folding it, then you probably have caiman/South American alligator. Its scales have a hard, almost bone like part, it will break and ruin the pieces if you try to fold it. Otherwise gator will fold like cowhide. To thick, or to tight can cause a split, so thin the backside as needed.
Regular contact cement works great.
Gator shouldn't be relied on for structural strength, layer it with calfskin or another suitable material to the desired finished thickness. Gator will tear between the scales if not backed, especially on highly stressed areas.
Hope this helps