How can I protect an electrical box embedded in rough masonry from weather?
Generally, it's a matter of providing a flat, smooth surface to apply any outlet cover to. This is a masonry/carpentry question, and not an electrical one. 

Your photos are so close I can't see the design context, but I'd fit a block of cedar or vinyl around the box. Shape it as needed on the back side to fit fairly well, then caulk and fasten it to the brick using a non-shrinking urethane or pure silicone. 

Extend the box to flush with the new surface, and mount a modern weather-resistant cover. 

On a related note, you can probably salvage the mount screw tabs by drilling out the broken-off screws with a very small bit. It's tricking to keep centered, but it can be done.