You can't decompress a ZIP file itself. Instead you would extract the contents of the ZIP file.
You should be able to right click on the ZIP file itself, and click on Extract All to extract all files from the ZIP file at once. You could also open the ZIP file, and drag and drop what you want to say your desktop to extract them from the ZIP file.
You may have to reset the associations for zip (compressed folders). http://www.vistax64.com/tutorials/65254-zip-file-support-restore.html
Another zip program may have been used in the past and now the default program no longer functions.
I was unable to see your screen shots, in lieu thereof, I saw small x.
It depends. Is the folder that the New World file in a zip file like in my screenshot above with the zipper on it? From your screenshots, the location looks more like a data folder for Minecraft.
What is the name of the zip file that you received from your first post? Could you post a screenshot of it unopened?