Did the Messenger of Allah keep a whip in his house? And what is the wisdom behind that?

If the purpose of placing the whip in his (peace be upon him) house was to keep it for lashing, then there is nothing to suggest that; in fact, searching for the whip for lashing indicates its absence. As for keeping it for travel or other purposes, there are indications in the Sunnah, such as his gesturing with his whip for people to move calmly.