We use a CAD table to cut all of our samples. This is a cutting table that uses a knife and cuts from a CAD file. You may be able find a packaging prototyping firm that could cut them for you, but it would be pretty pricey. I was thinking and have not done this before and can’t believe I am recommending it, but you could glue all you pieces up un-cut, draw your shape out on top and cut it with a band saw. Don’t know if it would work, but it sounds pretty reasonable.
As far as glue…Wood glue works great for these type of projects. Corrugated boxes are glue with hot glue or the industry term is “Hot Melt”. This could work, but with a normal house hold glue gun you will need to act fast as the glue dries quickly.