Hello, Mike! To answer your question, the demo contains most of the first area with the exception of a few locked doors, and there is currently no content beyond the two gates on the west and east (including the abandoned area you mentioned). The plot of the first chapter and all its required items are fully accessible, up until a cutscene that explicitly tells you the demo is over.
As for the fountain item, it can be found here (linked to avoid spoiling those who haven't played). You aren't the only person to have struggled with finding it - this seems to be a common roadblock among players, from what I've been told - so we'll look into how we can make this part easier or more intuitive in the full game.
Thanks for your feedback!