On second thought, and after reviewing a few sample motor speed circuits with various motor speed control IC's, I don't think the circuit has any place for a zener diode, it's probably just a flyback diode so you would test it the same way as a regular diode. Obviously, you would have to test a zener diode differently and map it to determine if it's working properly, and even then, you would need to know the specs before you can know unless you can identify it somehow.
The mysterious green device is probably a resistor although if it's green like that, it's probably a flameproof one or something. Now that I think about it, it seems that at approx 12v, a 47R resistor will allow approximately 250mA current flow @12 and some amount over 300mA @ 15v. I suppose that's about what you might want to limit current consumption at to the motor so perhaps it's a series current limiting resistor. If it failed short, it obviously wasn't limiting current which might've caused the controller to burn out or becomes damaged. Most cases, I see resistors fail by increasing in value, don't see them short as often though. As it appears that you were able to order a replacement IC, I say let's just count your blessings that they are still available and just wait for the replacement before tinkering more.
Here are some sample circuit diagrams I found online.