How is the generator controlled to give either as much power to meet the load or to supply the battery as well. Is the generator speed regulated in the model or does the model assume that the battery can be disconnected from the system?

Under load following, the generator produces just enough power to serve the load.  The generator does not deliberately charge the battery, so the batteries get charged only with excess renewable power.

Under cycle charging, each time the generator operates it operates as close to full power as possible (without causing excess electricity) and charges the battery with the surplus production.  There is a limit to the rate at which the battery bank can be charged, and the generator respects that limit.  The battery’s maximum charge power depends on its state of charge, its maximum charge rate, and its maximum charge power.  For details, please refer to the Help article “How HOMER Calculates the Maximum Battery Charge Power”.  In the cycle charging graph you sent, HOMER is governing the generator output with an eye to this maximum battery charge power.  As the battery state of charge increases, the maximum charge power decreases and the generator backs off.