Is the weather radar on and active during the whole flight?
Not necessarily; it is the pilot's prerogative to use the weather radar as required. If the weather is clear and the pilot decides that the weather radar display is going to take up valuable space in display, he/she can fly without the weather radar on. Of course, in cases like night flying, it is better to have it on during the entire flight, but it is the pilot's decision.

The regulatory requirements are for the weather radar to be operational only if potentially dangerous conditions are expected en route or for night flying. From CFR §135.175   Airborne weather radar equipment requirements:

(b) No person may begin a flight under IFR or night VFR conditions when current weather reports indicate that thunderstorms, or other potentially hazardous weather conditions that can be detected with airborne weather radar equipment, may reasonably be expected along the route to be flown, unless the airborne weather radar equipment required by paragraph (a) of this section is in satisfactory operating condition.

European regulations are also similar. From JAR-OPS 1.670 Airborne weather radar equipment:

(a) An operator shall not operate:
(i) (1) A pressurised aeroplane; ...
...
unless it is equipped with airborne weather radar equipment whenever  such an aeroplane is being operated at night or in instrument meteorological  conditions in areas where thunderstorms or other potentially hazardous  weather conditions, regarded as detectable with airborne weather radar,  may be expected to exist along the route.