Must a police car be inert in order to determine the speed of another vehicle?

First, it depends on what type of radar system is being used. Modern radar systems can be tied into the vehicle and compensate for the police vehicle's speed. So it CAN be used while the police car is moving.

Second, even if using an "old style" radar gun that doesn't compensate, there is no law or requirement that an officer needs a radar reading to issue a speeding ticket. Officers are trained to estimate speeds, including the speed of other vehicles while the officer is in motion. Radar guns make things a bit easier both for the officer and while in court, but they are not required. The officer can issue a citation based on his own estimate of your speed.

Police radar works fine in stationary mode as well as moving mode (just think even airliners use radar when they are moving as well).

Directional police radar allows the officer to clock cars in front of him either moving away or toward him and can do the same thing with cars behind him.

Police aren't giving you a speeding ticket because the radar told them to. Police are giving you a ticket because they visually estimated your speed and this estimation was confirmed by radar.

Radar is accurate to the 1/10 of a mile per hour. For your benefit the radar always rounds down. If the radar measures your speed at 99.9mph is rounds it down to 99mph

Yes, they can. They can match the speed of the patrol car with that of the speeder.