Powering off the router only resets the table of connected devices. All of the user settings (SSID, and key, etc.) are saved on the equivalent of a flash memory chip. There is a special switch on the routers for clearing this memory and truly reset the router.
The signal will be cut in half, but that is the signal - the amplitude of the data stream. Your router will be able to tune a signal within a certain range. Your signal would need to be very low for it to not be picked up. Rarely a problem.
Try the splitter if you find one, but in the meantime I suggest again trying the Ethernet cable from router to PC. It might yield clues.