How do I keep layered paper wings from curling once mounted on the wall?

The humidity theory is prolly right, and the desk vs wall difference makes sense. Flat on your desk, the strips had even moisture on both sides and could move freely. Mounted on a wall, the side facing the room dries out faster while the side against the wall stays damper, and that imbalance is exactly what makes layered paper curl. Thin strips with open space between them have almost no structure holding their shape, so they go wherever the moisture gradient pushes them. A hallway that gets humid in the mornings is honestly enough to do it on its own. You can slow the curl by sealing the paper. A couple of light coats of matte fixative or even diluted PVA brushed on both sides will reduce how much moisture the cardstock takes in. Do that before remounting, not after, or the shape locks in wrong. Switching the adhesive alone won't fix it though. PVA stays slightly flexible and spray mount is even more forgiving, so they'll let the strips breathe a little better than rigid hot glue, but they don't give the wing any backbone. The spine idea is the real solution. I've had good luck with a length of floral wire hot-glued along the underside of a layered piece, following the curve I want it to hold. A folded cardstock spar works too if you don't want metal showing. Glue it down while the wing is pinned in the position you actually want, then leave it under weight overnight before it goes back on the wall.

As for the shiny spots, yeah, hot glue does that through thin paper. It's mostly the adhesive saturating the fibers and turning translucent. Sometimes a very light dusting of the same fixative on top will knock the gloss back down, but honestly the spine will cover a lot of that area anyway. Mount the brackets lower on the wing where they're hidden and let the wire do the real work.

yeah the spine suggestion is the way to go, i did basicly the same thing on a layered dragon wing piece a couple years back. i used a bamboo skewer split lengthwise instead of wire, glued it along the underside with plain white glue while the wing was pinned to a foam board in the shape i wanted. held the curve perfectly even in my damp basement room. one thing i learned the hard way though, get the pins all the way out before you set the weight on it, i forgot one and it left a dent that showed forever.

Worth adding: even once you've sealed and re-mounted it, give the paper room to breathe behind the piece. I've hung layered strip work for years now, probably two dozen birds and fans, and the ones I floated a half inch off the wall with foam spacers stopped curling entirely. Flat against plaster is what traps the moisture imbalance in the first place. Also grab a cheap hygrometer for that hallway, mine reads 60-something percent every morning until noon and that alone explained half my warping back when I was guessing at causes.

One detail I'd add before you rebuild anything, since grain direction matters more than people expect with cardstock. If your thin strips were cut with the grain running lengthwise they'll flex along their length but curl badly across it, and the opposite cut behaves completely differently once moisture gets involved. I made a set of layered feathers last autumn and the ones cut the wrong way never sat flat no matter how long I pressed them, while the cross-grain strips stayed obedient. Worth checking your offcuts before you commit to a spine, because it might explain why some strips misbehave worse than others on the same wing.

For the shiny spots, a low-temp glue gun changed the game for me. I made a layered paper angelfish a while back and my old high-temp gun kept soaking through the thin stuff no matter how light I went. Swapped to a low-temp mini gun and the same cardstock barely went translucent at all, the glue sits more on top of the fibers instead of sinking in. Still visible up close if you look, but gone from across the room. Since yours is alredy built, I'd test it on a scrap strip from whatever you have left before committing to re-doing the brackets. Also, if you're flattening any of those curled strips first, parchment paper between layers and a stack of heavy books overnight, otherwise they'll stick to each other.