#1 yes, set to shore power breaker rating.
#2 charge light is on so your inverter is charging your batts, looks like they are at full charge to me. 12.5 to 12.8 volts.
#3 yes, need to turn on the inverter if you disconnect the shore power and want 110v. On some coaches when the gen set is started it will charge the coach batts. others you have to have the inverter powered and running to charge the batts. On my 01 DSDP 3852 the gen set charges coach batts directly when running from my 7k powertech gen set.
Last, I suggest that if you run your gen set to disconnect the shore power. Many electronic devices both on your coach and on shore power property that can be damaged by power surges. Example: solar charger, radio memories, tv controls, mirowave controls, AC controls.
May your black water hose never break!