Hello, thank you for playing Carpentry Story. Yeah, bug with mushrooms will be fixed soon, it' already known. What about the cave? I did not understood what happened to you in the cave. About save, it works fine for me. Did you unpacked the game before playing? If you play it from the zip archive, then save could not be created.
When you downloaded the game, you need to unpack the archive (right click and extract here) to have the game folder unpacked/extracted. When you do the save in-game, minimize the game and check the game folder for the save file, it should be in ".../www/save/fileX.rpgsave", where "X" is 1,2,3,etc depending on the slot you've saved.
I'll try to recreate the cave bug and hopefully this can be fixed, too.