Everyone wants a tactical advantage over the enemy. There was always a fear of falling behind the enemy so military R&D went into overdrive during WWII. If you weren't constantly trying to make your aircraft faster/lighter/more durable/better armed you were probably going to be eclipsed by your adversary sooner or later.

I don't have any sources of how failures were reported, but I'm sure every fighter pilot wanted the next generation to have a more powerful engine, less drag, better armor, and more guns. In a dogfight, speed is life for a pilot. Battle footage could have been utilized to back-up claims made by other pilots about performance during air battles.

The best example I can recall happened during the Battle of Britain with dogfighting between Spitfires and BF-109s. Spitfires had engines with carburetors while the 109s had direct fuel injection. The German pilots realized that if you could get the British pilots to chase them at a certain angle of attack, the g-forces would starve the Spitfire engines of fuel, causing them to stall. The British pilots would then be sitting ducks trying to restart the engine during the stall, or they would just have to bail out or crash. The British realized this and soon all their aircraft were fitted with direct-injection engines.

In terms of training, I know the newest iterations would often be pressed into frontline service so fresh pilots would sometimes train in older aircraft. You don't see a huge difference in aircraft handling and speed until after the war, (jet age) so pilots would sometimes have to compensate during a sortie, but it really depends on which nation you are referring to.