Two power sources in a single gang switch box
Just looking at the connections in the switch box and the voltage readings on the separated wires, it appears that cable 3 (lower right) recieves power from the service panel, cable 2 (lower left) supplies power to adjacent rooms, and cable 1 (upper left) runs to the light fixture. - What is completely surprising is the actions of the two circuit breakers. 

Perhaps you should begin by reconnecting everything. Then, switch off the circuit breakers individually and in combination, and note which switches and lights stop working in each case. If you find anything that goes dark when either breaker is off, or anything that goes dark only when both breakers are off, then your house is dangerously miswired and you need to fix it before installing anything new.