Nope, you need to open mysterious chests from the event itself.
I suggest to quickly open them once after you got some together with those you already have in the inventory, from Marco, purchases etc. Just incase.
This event might develop bugs and exploit.
No, but I am keeping some on my base in case they do. It appears that 3 is the max that will sit there though - I had three, got a new one, and the oldest one had disappeared. Either that, or it moved the oldest one to a new spot.