A thick outline isn't necessary. The thick outlines of the characters are designed to make them stand out from the background, so you can understand why they wouldn't be as important for the backgrounds themselves. A thin outline wouldn't be a bad idea, though.
It's good to see you working on the graphics for this stage, since I know a lot of people are really looking forward to some reference to An Untitled Story.
The graphics need to have a more cell-shaded appearance, though. Such smooth blending doesn't fit the game's style, nor does it suit pixel art's strengths. You should not need to use more than 3-4 shades for the entire stone structure. You ought to up the contrast, too, so the details really come out.
The stage needs to be at least 384 pixels tall, but should preferably be a bit taller. Give it a higher ceiling, and a bit more space for the floor. It looks like you should be able to stand on the structure's roof, so there needs to be enough space to see any action going on up there.
As a minor note, the pots look like they're viewed from an isometric angle, rather than side view.
Finally, this isn't strictly necessary, but it would be nice if the level stuck to a 16x16 grid. Jumping is balanced around this assumption.
Again, it's really good to see some progress on this. Keep up the good work.