The key for the rusty chest is created after brewing and using a specific potion. The potion is very literal in its effect. You have in fact actually picked up the key before. The Lilyth is the mushroom that automatically ends the dream in a nightmare unless you have first brewed and used a specific potion. You haven’t softlocked yourself. It doesn’t involve the Void Forest, however having True Sight in there makes it much easier to escape with the key. You don’t need to start in the Lab and I highly recommend you don’t unless you are specifically trying to bring a Lvl4 Card to the lab.
The bookshelf requires a book that is located in the basement where you found Yuri. Interacting with that shelf drops a red book provided you have painted the blank book at least once. You do not have to repaint the blank book to spawn the red book. As far as I know there are three sets of important numbers. The blank book is not one of them. Two are static and the last is randomized every new game.
Spoilers
I forget where exactly it is first mentioned but you can search the school library for Festival of Life to find out the general requirements for making it rain. My notes say that you can learn about the Festival of Life and a set of numbers by talking to a specific ghost tree located by the fairy circle of mushrooms.
The Crusty key can be transformed into the rusty key. It requires a recipe mentioned of Page 15 which I believe is buried beneath the tree that want a Lilyth. You do not need to give it one before you can dig it up. It is also the tree that gives you the Crusty Key when you search its roots at a cost of the blue resource.
The recipe to safely handle the Lilyth is located in the area you can reach by interacting with the green portal that will appear between the two small trees directly below the mushroom circle when it is raining. There will also be a book telling you one of the alternative methods to making it rain although it does come with a minor downside. (A specific mushroom will not appear.)