A converter takes 120V AC power in and produces 12V DC power to operate 12V devices and charge the 12V battery.
An inverter takes 12V DC power in and produces 120V AC power out to operate 120V devices (from energy stored in the 12V battery, usually).
An inverter/charger is a dual-function unit that combines the functions of an inverter and a converter.
The generator produces 120V power directly. No inverter is needed when using the generator to have 120V power, but the converter is operating as there's 120V power and it powers the 12V system.
RV generators generally do need 12V power for the starter motor and often for the exciter and control circuitry, but don't let that confuse you as to their overall purpose: they change gasoline (or propane or diesel fuel) into 120V AC power.
Yes, it is possible to add an inverter to your setup if you don't have one--or, for that matter, even if you already do--to use 120V things without needing the generator running or shore power connected. There are several setups that can be done with varying levels of installation effort and convenience of use. There are also some improper setups that could be dangerous, as with just about any electrical device installation.