You can leave your solar plugged in while the genset is running and it won't hurt anything, but as Gary said, you won't notice any faster charging times. And yes, your converter will power both the 12 volt house systems and the charger while the generator is running, so your batteries won't drain - they'll charge. How many amps your generator will put into your batteries depends on the charging stage (bulk, float etc.) the output capability of the charger and other things in the coach that are running. The converter shares its output with the charging circuit and other things that require power.
When our house-batteries are below about 85% SOC, the battery monitor indicates that the charger's output in the bulk charge stage is 150 amps. When the batteries are fully charged, however, the charger's output indicates only 1 or 2 amps (if nothing else is on.) Your charger's max output is probably less, but your battery monitor will display how many amps it's putting into the batteries at any given time.
Kev