For the desk, exit the Shelter (the hub area with all of the shops) through the south east exit. You'll go down in a vertical room and west through a long room. You'll come across a room where if you fall through the gaps in the foliage you'll end up in the area below with no way back up except to go through the Shelter again. Instead of falling, keep heading west. You'll need the double jump or a jump with the bicycle equipped.
The "black box" is the trash bin. That is in the dense woods (the forest area with the acid rain). If you enter from Block City, it's a straight path with no detours all the way until the first save point. The trash bin is in the 9th room. It is the vertical room just before the fork in the road that leads to a save point. It'll be in the upper left corner. You'll need the double jump to reach it.
The Carpet is in the Snow Mountain area. If you make your way all the way to the room with the vending machine, go south one room. Then drop down the gap immediately to your left. Allow yourself to continue falling through another gap. You'll end up in a little corridor that leads west. Follow it and jump over the little gap at the end. The Carpet will be in the next room.
If that doesn't make sense, let me know. I have a world map that I'm translating that will be included along with my translation of the game. I just don't have access to it right this moment since I'm not on my usual computer.
If you find them, defeat the last boss (if you've already defeated it, defeat it again) and you'll unlock Houchou Shoujo Mode!