First, the Mongol cavalry is more of a skirmisher than an MBT. In any situation where they have to fight heavy cavalry head on, they suffered heavy casualties. Their specialty is hit-and-run and ambush tactics.
Secondly, the reason Yue Fei is legendary is because he was able to achieve his victories DESPITE a corrupt, ineffectual, and ultimately even hostile government. Had the Song emperor not recalled Yue Fei, it is quite possible that he would've rewrote the history of China.
Not exactly, although Mongol calvary is famous for its hit-and-run and ambush tactics with light calvary, it did have well-armored heavy calvary and use them wisely. In many battles particuliarly those against European lords, the Mongol army delivered their final lethal blow with their heavy calvary after they managed to wear down the European army´s momentum by eliminating its mobile units.