Regarding the "black box" circled, one wire is attached to the output, the other is bare. It looks like it is connected to the base of the relay. If it was part of the relay, there would be a mechaincal connection as well. Since there is a resistor also, my first guess is that it is a time delay or flashback arrestor for the coil of the relay. The relay is otherwise self contained. When it is powered on or off, a click will be heard, one(1) time. If repeated clicking is heard, it is indicative of a problem.
The motor capacitor is the large, black, circular device in the upper left of the photo. I honestly am not familiar enough with the brand to say for sure. But know that I am NOT very impressed with the workmanship of the device. It has the appearance of some jack leg "am damateur" construction from a magazine article. As a real elecrrician, I would have to say that if you paid more than the cost of a soda (or beer) then you paid too much.
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