Hi 1L243,
My solar and all forms of charging play well together. So I would go with #1.
My preference is to shut everything down when I switch to using the inverter. The transfer switch I have is internal to the inverter/charger and I'd hate to burn the contacts, so switching at zero load is best. Items don't need to be unplugged--but I do switch them off, other than the microwave clock. My default for the converter is unplugged.
I use dawn dish soap sparingly with lots of water and a soft microfiber cloth. Mostly I don't clean them at all.