Yeah, the comic could really use a fourth panel showing just how absurd it is for war horses to climb ladders.

The 'wall' was constructed centuries prior to the Golden Horde, which existed in western asia and did not threaten China. There were centuries of construction and maintenance but most of the prominent portions of the system was built/upgraded after the Mongols by the Ming, who overthrew the Yuan dynasty, (which were the Mongols).

The one battle the Mongols fought at the wall they just went through the open gate. In one instance they just paid the guards.