Any OEM install should not present any risk to leave the inverter on.
Reasons to leave it on would be to power the fridge if the utility power goes out or to be sure you don't forget as you depart.
Yes it is best to turn off in storage. Good chance the inverter may bypass the disconnect switch, might be worth checking but still best to power it down with the intended inverter power switch.
Yes the solar should charge regardless of the battery disconnect switch position. If you service the battery it is best to disconnect the panels from the controller before you disconnect the controller from the battery.