On the facing, you can use any adhesive that you would with card stock. If you want, hot melt works too.
The foam core will be disolved by solvent based adhesives - such as most spray on types.
Corners - different people use different techniques, give each a try and see what works for you. If you are covering it up with brick paper or similar, I'd just use a butt joint. A corner rienforcement doesn't hurt either, like ΒΌ" square balsa.