Sometimes greys will fixate on one particular toy, he may be "in love" with it and getting down under the cage might mean he is thinking of nesting, you can try putting something down there that will discourage him from going there. My grey will occasionally go under her cage but she seldom spends much time there but if he is spending all his time there then it doesn't mean he is sick, as long as he is eating and defecating ok then you need not worry about that.
Most greys love bells, mine has a heavy duty one that she likes to bang around especially when she is a little pissed off, she will take her frustrations out on that toy but they all have their favorites, I see no harm in him wanting to destroy cardboard, its cheap and wood toys are great.
The wing feathers will grow back several at a time with the same ones on each side being lost and replaced, I would imagine a lot of them have grown back since he was last clipped in May, hopefully you will consider leaving him flighted, you do have to be extra careful of open windows and doors but to hear the flapping of wings is music to our ears.
Destroying things is what they do, they need things they can tear apart to keep them busy when they are confined to their cage, don't want them to get bored for then they make start working on themselves, I look at it like this, a toy destroyed is a sign of a happy bird.