Hot glue would not work well for fabric. Fabric glue would work to make the hat without sewing, though the pleats may be a little tricky, and the hat would not be as sturdy.
Yes, but it's not really worth it since you will be adding shaving cream. The lotion, shaving cream, and foaming hand wash will make the glue look white, instead of...
Use bonding web, or iron-on fabric glue, to hold the patch in place. However, since that isn't a permanent solution, you still need to stitch the patch down.