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The sensor itself is out of the scope of this technical support team, I’ll recommend you to ask the manufacturer for that kind of information (expected operation time, continuous working time).
It seems that the FC-37 is the most widely used rain sensor. Many projects involving the FC-37 have been shared in the web, I found easy to follow code in this two sites
Due to the structure of the Arduino UNO and the Edison, the ports are compatible, keep in mind that I can’t guarantee library compatibility, but you can always program/modify your library in case an issue emerges.
I did some research, but unfortunately, there is not a community thread that mentions anything similar to your project, so I can’t suggest alternative rain sensors for the Edison. What I did found was a rainwater module, similar to the one you suggest, connected to a Galileo board http://www.build-electronic-circuits.com/intel-galileo-board-rain-and-moist/.
The Rain Bird CPRSDBEX Wired Rain Sensor is designed to work with 24 V (alternate current) sprinkler controllers (residential and commercial), and there is not a board (Arduino or Edison) that can generate that output. That’s why you haven’t found any information regarding that setup.