Does the ISS keep a standing watch with one person always up and awake and monitoring while the others sleep, or do all the astronauts sleep and work and they all get alerted in case of issues beyond the normal scheduled stuff?
It definitely does, and not just one person - except these people are ground-based.
The astronaut crew is only a small fraction of the number of people operating, controlling and maintaining ISS. While astronauts operate in shifts, and usually (though not always) at the same time, periods when everyone on the station is asleep aren't anything infrequent. Especially after long EVA, where two astronauts work outside the station, while others take care of the robotic arm, systems, maintenance, and aid these on EVA in preparations and return, you'll have everyone asleep after a long period of joint activity.
Meanwhile, the control center operates 24/7, and save for brief periods where ISS is outside reach of all ground stations, they perform maneuvers, perform IT maintenance, control experiments that are remotely operable, monitor station status, apply regulation of life support systems, and so forth, and so on. There's way too much work just for the astronaut crew - the astronauts only do what can't be done remotely, or doing what remotely would be impossible, dangerous or troublesome, e.g. due to transmission delay (ping), or lack of remote control infrastructure.