Do police use automatic license plate readers?

Can police officers randomly run your plates using ALPR?

The simple one-word answer is Yes. A police officer can run your place if you are suspected of criminal activity (in the USA). This could differ depending on where you are in the world. There have been court rulings saying that an office can check a license plate since there are no privacy laws about the vehicles license plate itself.  

Can anyone just lookup a license plate number online?

If you are in the USA, then every car has a VIN number and you can easy look up the VIN number online. There are many sites you can find this info. Just Google free license plate lookup. (yeah it is really that simple to look up a car history) 

What is LPR’s becoming so popular in police departments?