I can't see an easy way to get this little box (circled in yellow in the picture) open to see what's inside. It's connected to the fender on the left side of this picture by a single bolt, and on the right, it connects to the air intake hose. So what is it there for?

 

It's a noise-cancelling chamber for the air intake. It really has no other purpose but to eliminate some intake noise. It is a molded item, so you arent supposed to be able to open it.

 

So if I remove it and cover the hole in the hose, how would it change the sound? Better, worse, throaty, whiney?

 

probably more throaty, but I havent seen it done on a 3.5, so I cant say with any certainty.

If you do, make sure you seal the tube well, as anything that gets in bypasses the air filter. Also make sure your plug cant get sucked in, as it could get caught in the throttle body and hold the throttle open.