Who assigns an arriving aircraft a runway and runway exit to use?
The tower gives the ultimate clearance to land, though the approach controller (if one is available) assigns a specific approach to a runway in the approach clearance. I'm not sure about the game, however in real life if the runway was too short to land on, the pilot would report being &quot;unable&quot; to use that runway and would be assigned another runway instead. It is not the responsibility of ATC to ensure that the runway is long enough to land on.

Unless there is a NOTAM otherwise noting that a particular exit is unavailable for use, the pilot can slow down at whatever rate to vacate at any particular exit. However, once they've slowed down to taxi speed they should vacate as soon as practicable. The tower controller may request that a pilot vacates at a specific exit, but if they are unable to they will go for another exit.
After vacating the runway, the pilot notifies the tower that they are clear of the runway at a particular taxiway, and then will be (depending on the airport) be asked to switch to ground or be given taxi instructions.