Perhaps not true on bronze, but since i've been looking, there is no winter cup, and there are two summer cups. Winning both summer cups does not give me the winter cup trophy though.
After beating the phantom in the little giant battle, when he comes back to challenge me, accepting results in a frozen "do you want to practice" screen.
TIP: A bug that can work in your favor- using "in training" items with experience actually yields double the experience given, it is added on to the usual training, and then again in the bonus portion.
The boss monsters come back to challenge you and they are weaker? So he traveled all around the world and learned to have much less power and health? The only challenging monsters left are in the training section, but they don't give you any experience or money, just a couple of stat points.
 "Chuck the Sheep"){kind=icon}
 "Monster Arena"){kind=icon}
 "Reincarnation: The Final Happy Hour"){kind=icon}
 "The End"){kind=icon}