Even if you have an old mercury switch thermostat, it's still really easy to swap. And if you break the old thermostat, they're really cheap. I kept my old one and it's just sitting in the pantry in a box with the locks in switched out for August Locks, and the original door bell I swapped lol.

You can't make zone control where it doesn't already exist without a bunch of work that your landlord would absolutely need to handle. But Nest has wireless temp sensors you can put in a room and I believe set your temp off that specific sensor.

If your landlord isn't garbage they'd likely do it for you. Mine offered but I didn't want to waste his time.