Thanks for checking out the game!
Can you describe the issue you are having with the equipment menu more? You are the first to mention it not working. E key opens and closes it if all is working properly.
Currently the only saving that exists is the results of your runs which can be viewed by looking at the grave stones. It occurs any time you die or exit the game via the pause menu. As of right now you can't save while you are in the dungeon. If it is something that gets asked for enough I will look into it in a future update, whenever I have time to dedicate to this protect.