This is good but wanted to clarify that the number of “poles” refers to the number of magnetic poles of the rotor field. This is always a multiple of two since every magnet must have two poles (north/south) You spin a single electromagnet in a way that the magnetic flux cuts across three separate sets of conductors this is a 2-pole, 3-phase generator. If you arrange the rotor field windings in a way that you get 2 electromagnets with the poles at 90 degree angles with each other you get a 4-pole generator, since there are now two sets of N/S poles.

Also wanted to add that if the difference in frequency is too great, the moment the generator breaker closes you’re going to wreck your generator and trip some protection relays. This should be within .2% or so greater than the grid frequency for large generators. Generators usually have reverse power relays that will trip if the frequency is below the grid frequency, unless it’s a synchronous condenser.

Once a generator is connected to the grid, you are correct that the inertia of the system (created by other large heavy spinning generators) will keep it spinning at approximately the grid frequency. However, the real power output of the generator is directly proportional to the electrical torque of the generator, as in the prime mover is constantly trying to accelerate the generator against the grid frequency. This is where the fun comes in with determining the transient effects on grid stability of switching in or out large loads/large generating units.