If you can post a few pictures of the underside of the stairs that'll be very helpful. As you suggested, maintaining 6'-8" headroom is critical.
Good solid advice there! Code doesn't say you can't hang rock (or framing) directly to the stairs, I don't think it'll be a problem as long as the stairs are rock solid.Can't say as to code, but every one I've hung has had the framing members (generally 2"x4" laid flat, not on edge) fastened directly to the staircase however is was possible. The stairs are freestanding and need no structural support. The framing underneath is just to hold up the drywall. The drywall will have to be hung with 2 layers of 1/4" to allow it to bend and bow around the radius of the stairs. Having the widest part of the 2"x to fasten to makes it much easier to fasten as the pieces are difficult to get an exact measurement for. It's easiest to cut them on the large side and "whittle" them into place. The more "nailers" and the closer together, the better off you'll be. Be sure adhesive is used to minimize the amount of fasteners......