Bless your heart! The only thing that comes to my mind right now is placing shelves on the walls in each room. Not book shelf furniture, but shelves on the wall that you can vary the height of and the design on the walls. I would stagger these shelves a bit to add variety to the space. Place the stuffed animals on them and place the puzzle boxes. Maybe frame a few completed puzzles to place either on the shelves or near them.

Displaying collections might have to come down to "rotating stock". This is hard enough in design...with people with obsessive issues (autists have plenty of obsessive concerns) this could get a bit sticky. My little autist has very limited collections and I've managed to keep it that way (mine is 9 years old = give it time and his collections shall become as large as yours).

If you can "offer" choices (my little autist has grown up with "this or that" concepts so it would work in our house hold) to allow "X" amount of product on display at a time. That way you can try to keep things down to a dull roar.

Cubby holes for displaying a small number of stuffies (15-25 at a time) would help. The puzzle display (have you mounted them/glued them?) would have the same options...just a bigger display case.

This way you can "offer" a set day of the month where the change over happens and they can plan for that. You can pick "themes" for the displays like colour, design or "species" (stuffies). You get the idea. It is an option in the neurotypical world that may or may not work in your world. It all depends on the rigidity/flexibility of the guys in your life.

I like the idea of offering choices. I have a daughter with mild delay and offering this or that got me through alot with her. If not, we would have had 1000 Zac Efron posters, Harry Potter toys and endless Harry Stiles tshirts in her room. Perhaps also having a hanging net that can hold the stuffed animals could be a good way to still have them visible to your sons eye, but keep it less cluttered. He probably finds the stuffed toys soothing and anchors him.