What questions MUST we ask the landlord during a room/apartment viewing?

Always explicitly ask, "What other fees would I need to pay in order to live here, outside of rent?" Often there will be shit like amenity fees and parking fees that you otherwise won't know about until the day you get your lease. Very annoying.

Ask about nearby grocery stores and restaurants. You can find that on google maps, but often the landlord will notice something you don't.