Satellite Wifi Snooper with PoE
Yes, you can (ab)use the unused pair of wires in cat5e to provide power. A DC power supply would not interfere much with an Ethernet connection. You can buy passive POE injectors that take care of the wiring for you. Or an active POE to 5V adapter if you have real POE available. 

But you need to calculate the voltage drop. Cat5e is 22 to 26 AWG wire, and a 30 meter one way run will have a significant voltage drop. Using a calculator and 24 awg at 180 feet round trip. A 1A load at 6V source will mean a 4.5V drop at the device end. Not very good. You would need a higher source voltage, and due to the varying load, a voltage regulator at the device end. 12V with a 12V to 5V car usb adapter would work lovely.