Can't speak for the switch but yes the generator or utility power drives the converter to charge the battery. Yes it always draws some 120v power when available. More power with low battery, less power with full charged battery.
Two hours is plenty with a good multistate converter that I suspect you do not have. Post the converter or power distribution model# for best suggestions.
Solar will charge a low battery but depending on wattage it may take an extended time and need long sunny days. If you put 200w panel up there it should charge quite well.
On edit:
PD9245 is actually a very good charging converter. But verify you are getting 14.4 volts on the battery terminals when you expect fast charging.