Is the climb rate different for short-haul flight and long-haul flight?
Different climb performance is indeed indirectly linked to the length of the trip, but this has to do with the number of engines. All airliner's engines are sized to support flying with an engine failure right after take-off. Twin-engined aircraft enjoy slightly less stringent rules, but will basically carry twice the thrust that they need to stay aloft, so more excess power is available than in four-engined aircraft, which have only 33% excess thrust.

Generally, the best performance is achieved when running the engines at their maximum continuous thrust setting and when climbing as fast as possible. That is the reason why an A-320 or a Boeing 737 will climb faster than an Airbus A-340 or a Boeing 747. On descent, the number of engines is less relevant, so the descent speeds of all types are similar.

The link between engine numbers and range was strongest in the early jet age; now it is less clear-cut with long-range airplanes like the Boeing 777-200 LR or short-range airplanes like the BAe 146.