Hi there! There are quite a few puzzles to solve and ways to progress, it's hinted at in the game's description on it's page. The game does keep going once you die, if you restart it and choose to continue, everything you did in the apartment, like putting an item into that box, stays the way it was even if you close the game (or the game closes on you). I like to leave that stuff for the player to figure out, though, so I can understand your frustration if you didn't know cause not a lot of games put enough faith in their players to do that these days.
You're absolutely right about there not being any way to quit the game in fullscreen mode, though! I only ever tested it windowed where there's a little x to close it. Guess I gotta implement that, or pretend you're in it for good once you decide to open this window into another world. spoooooky~