Must a Boss use their Level Up ability?
According to the rules, your boss's "Level Up" ability is triggered when your dungeon is 5 rooms in size. "Triggering" implies necessity. So yes, the only way to avoid levelling up is by not building a fifth room.

Secondly, and more to your concern... according to the Advanced Rules (see the entry for Robobo in the glossary using the link above), using Robobo's level up does not trigger the "destroy this Room" ability for the destroyed room.