If a car is running rich over a long period, the sensor may become plugged up or even destroyed. Just shorting out the sensor output wire will not usually hurt the sensor. This simply grounds the output voltage to zero. Once the wiring is repaired, the circuit operates normally. A cheap voltmeter will not be accurate, but probably will not cause any damage.
Resistance measurements send voltage into a circuit and check the amount returning.
An oxygen sensor is a chemical generator. It is constantly making a comparison between the oxygen inside the exhaust manifold and air outside the engine. If this comparison shows little or no oxygen in the exhaust manifold, a voltage is generated. The output of the sensor is usually between 0 and 1.1 volts. All spark combustion engines need the proper air fuel ratio to operate correctly.