Thanks so much for checking it out!
I'm sorry for the bug. I hadn't seen an issue come up with a player's door opening early, either in testing or in playthroughs I've seen. I couldn't replicate it but I believe there was a rare situation where one of the puzzles could be double-counted due to network delay. I've now updated the game so that there is no way that any puzzles could be double-counted, so I believe that the bug is resolved.
You are correct that there are 5 objectives that must be completed for the door to open. The beginnings of some puzzles are less visible than others--if you are stuck, I would recommend that you focus on objects that one player hasn't had to use in a puzzle yet (Or hasn't had to communicate to their partner), which might help you consider new ways to interact within certain rooms.
Thanks again for playing!