I think you have some good ideas except for what you are planning on storing in there...I'm thinking ammo and handguns and some more ammo and some loaded mags, some more ammo, and maybe a few long guns in the closet.

I would go for a center support or opt for better quality wood than plywood.

Going to the ceiling depends on how tall you are and if you want to reach way up. You can also cut it off at around 6-7' and put something on top be it more storage materials like ammo, or decoration