How do I launch a file with the extension .darwin.amd64? How do I launch or open a file with the extension .darwin.amd64?

Double clicking it results in: "There is no application available to open the document..."
If there is no file extension, then it is most likely Mach-O file format. This kind of files without file extension can be executed within terminal by dragging the file to the terminal window and pressing enter. Another way is to use terminal to cd into the directory containing the file, and executing ./filename.darwin.amd64, of course replace filename with your actual filename.