In what scenarios would one want to maximize or minimise rate of climb in an aircraft?
Dogfight is about energy management. You care less about climbing than about being able to turn your nose and weapons into the right direction as quickly as possible without falling out of the sky.

Maximum climb would be required in those scenarios:


Fly as efficiently as possible. Flying higher means flying faster and in colder air, so the engines are more efficient. Note that todays airliners climb as fast or faster than fighter aircraft in WW&nbsp;II.
Evade ground threats. After delivering its ordnance, an attack aircraft would desire to get away from hostile fire as quickly as possible. Sometimes this is achieved by flying with the best climb angle, though.
Break off combat: If you know you can climb faster than your opponent, you can choose whether you fight or get away, a luxury the other guy does not have.
Intercept: If you were ordered to intercept an unknown aircraft flying high above, maximum climb speed is helpful. If you need to be fast as well, your optimum climb profile is that of highest energy gain, though.


I interpret minimum rate of climb as maximum descent rate. Typical scenarios are:


Pressure loss in an airliner, when higher density air should be reached in order to continue the flight safely.
Dropping parachutists: If the aircraft lands ahead of the skydivers, it can fly more often and earn more revenue.
Extreme case: Dive bombing. Actually, most dive bombers employed air brakes to limit the speed in a dive.