Final answer:
There is no historical record of Genghis Khan using cats or swallows in sieges. However, his army did engage in biological warfare by catapulting diseased corpses into cities to break enemy resistance.
Step-by-step explanation:
The use of animals such as cats and swallows by Genghis Khan and the Mongol army to capture a city is not recorded in historical texts. However, the Mongols were known for their innovative and ruthless military tactics. An instance involving biological warfare was when they catapulted infected corpses into a city to spread disease among the city's defenders, weakening and eventually leading to the surrender of the city. This method of warfare was used during their conquests after Genghis Khan's death. It should be noted that the example given in the question could be a hypothetical scenario as there is no historical record of such an event involving 1000 cats and 10,000 swallows.