Yes, but you'll want one per floor.

Yes, it will work, but the robot can’t climb stairs. You have two real options: buy one robot and move it floor to floor, or buy a robot for each level.

Moving one robot works fine if you’re disciplined. You pick it up, carry it upstairs, plug it in. The mapping software usually handles multiple floors, so it knows where it is after a few runs. It’s cheaper, but you have to remember to move it.

Getting a robot for each floor is the set-it-and-forget-it approach. Each one stays on its own level and docks. That’s the better experience if you hate chores and have the budget. Most people eventually go this route after the first week of carrying the damn thing up and down.

As for stairs themselves: the robot will stop at the edge if it has a cliff sensor (which they all do). No, it won’t fall off. Just don’t leave the sensor dirty.

You don’t need a fancy multi-floor mapping system unless you’re moving the robot daily. Most modern robovacs handle it fine.