Did the violence the mongols brought somehow normalize/legitimize the use of violence in these cultures?

Are you asking if Mongols brought violence to these people? If so, then no. There was plenty of warfare between states, especially in China and Iran, that brought extreme loss in populations.

The Mongols' ruthlessness was partly a psychological warface tactic that worked wonders for them at the time. Described as the Devil's horsemen, it was easier for their enemies to surrender after hearing a couple of horrifying stories of total annihilation of different sub cultures.