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.

Just as an aside from the basics, when looking at apartments in apartment complexes or buildings, I’d actually try to catch a few tenants who were milling about to ask how they liked living there, positives, negatives, etc. to get more of a feel for the place.

Yes, ask about other fees.
The rent price may be fine but once they tack on extra fees, it maybe be $100+ more per month than you expected

Water and electricity billing. Waste management. If neighbors are kind.

What will you do when the workers rise up?

Does he have any hidden cameras?

If you can do any alterations and to what extent.
What to do in case something breaks down.
Extra fees and possible unexpected expenses you can have there.
Ask about humidity and whether you'll need a humidifier to prevent mold.

Depends but I'd suggest doing a video on the day you move in documenting the state of every room and damages so you can get your deposit back.

Are they nosy and overbearing?

Get them talking about damage previous tenants have done to the place. There's always that one person who treats a rental house or apartment like it's a punching bag that you live in.

What justifies you charging $2,500 a month for nothing more than a closet?