I double-click the .zip, and the resultant folder shows up here: 
Annoying, but eh. Is there any way to fix this? Similarly, anything downloaded to the folder shows up in the top left, piling on top of anything else that might be there. This ends up in a giant stack of mess that's annoying and cluttered. How do I fix such problems?
You could change the "view" of the ~/Downloads folder to something other than "as icons" from the Finder menu "View". In "list" or "column" mode, they would not be on top of each other.
Alternatively, under the "View"->"Show View Options" you could set it to "Sort By" -> "Snap to Grid" and all new icons would be nicely arranged it a grid. Additionally "Arrange by" -> "Name" would put things in alphabetical order, and not all on top of each other.
Change sort by to something other than none or snap to grid, e.g. change it to name or creation date.
(I am new to macos and expected "snap to grid" to actually keep the icons in a grid but with this option set it behaves just as in your screenshot. I don't really understand why, but using another sorting option fixes it, so whatever)
