They're not direct driving the screw with steam power from the reactor heat-exchange, surely.
You've probably seen aircraft carrier decks covered in smoke. This is because the Nimitz's catapult system is directly steam powered, service steam is widely used on the class. I can't find a citation for this now (this is from memory) but I saw some documentary on it once where it was mentioned in passing that even the washing machines on the ship were steam powered. There's service steam everywhere on the Nimitz.
Having steam power everywhere introduces a lot of complexities, which is why the Navy's moving away from it with their new carrier class[4]. The Gerald R. Ford-class does away with service steam in favor of more powerful electric generators. There'll be no service steam, just steam to drive the electric turbines.
1. http://www.nimitz.navy.mil/FACTS.html
2. https://en.wikipedia.org/wiki/Nimitz-class_aircraft_carrier#...
3. https://www.marinelink.com/news/dresserrand-propulsion302604
4. https://en.wikipedia.org/wiki/Gerald_R._Ford-class_aircraft_...
https://en.wikipedia.org/wiki/Electromagnetic_Aircraft_Launc...
Basically they spin up flywheels that then dump their kinetic energy back into electric power as the catapult is released.
And it allows them to finely adjust the speed of the catapult depending on the kind of aircraft they are launching.
Yes, they are, for the simple reason that it's more efficient. This is beginning to give way to a series electrical design, but that's for reasons of increased stealth and other concerns, at the cost of overall thermodynamic efficiency.
On all existing nuclear carriers steam is used for the catapults (and this is much more demanding than many people are aware of). The Ford class is developing electromagnetic catapults, which are also proving to be demanding to perfect.