Duck Game – How to Customize Your Room

It's honestly difficult to say. Looking at this Steam Community discussion, it's difficult to tell the exact conditions required to unlock them.

One post there, though, describes in detail how they unlock.

They unlock weirdly. To unlock more after the first 3 you need everything on gold and at least 5 platinums from those. After that you'l unlock a bonus challenge map of the first obstacle course. Do that and you'l get another arcade machine. From them you just need to beat each level in all of the available arcade machines to unlock more, you don't need any gold or platinum medals. x

What I would personally recommend is to try going through each unlocked arcade machine through a more systematic approach. Try reaching for gold/platinum in the earlier challenges before moving on to the next ones. From there, you may be able to unlock the rest of the arcade machines.

How long does it store the item?

Does the box save across matches?

It persists through the entire session. It also persists across sessions if the host decides to play another round while displaying the trophy board (I don't know what happens if you go back to the lobby first). It doesn't persist if you disconnect.

If another player puts a different item in, does it change it for you? Or will it give you the same item you put in?

The save block is local to your player only; items that opponents save do not affect the item you have saved.