A synchronous generator can be operated over some range of speeds. The torque characteristic would be a vertical line at the driven speed. The maximum continuous torque would change somewhat depending on the limits for voltage and power. The excitation control would need to be designed to accommodate the desired speed range.
It is usual to concentrate on fixed frequency operation when studying generator fundamentals. When studying generators for use with wind turbines, the variable-speed characteristics are considered.
Re: Additional questions added in comments:
The fundamentals of connecting two AC sources to the same network should explain why they need to be synchronized in the required manner. Texts have sections covering the connection of a synchronous generator to an infinite bus and connecting two generators to each other. Vector diagrams, equivalent circuits and example calculations are provided. The material explains what needs to be done to assure that the generators are operated only with their rated parameters. There may not be much material available that studies in detail what would happen inside a generator if operation outside of the rated parameters is attempted. It is possible that some technical papers have been written on the subject. Google scholar may be able to find something.
Quote from a technical paper:
When an off-line generator is energized while on turning gear or coasting to a stop, it behaves as an induction motor and can be damaged within a few seconds. During three-phase energization at a standstill, a rotating flux at synchronous frequency is induced in the generator rotor. The resulting rotor current is forced into paths in the rotor body, similar to those rotor current paths for negative-sequence stator currents during generator single-phasing. Rapid rotor heating, and damage to the rotor will occur. The machine impedance during this high-slip interval is equivalent to the generator negative-sequence reactance.
C. J. Mozina, "Lessons learned from generator tripping events at industrial facilities," 2008 55th IEEE Petroleum and Chemical Industry Technical Conference, Cincinnati, OH, 2008, pp. 1-8.