Yes.
Yes, but it does depend upon your inverter.
If you don't want to split your circuits you can either, not bother using the essential output (and have the lights out when the mains is off) or get an off-grid hybrid with sufficient oomph to supply your whole house and forgo exporting.
Note:- There are off-grid hybrids that can export, but they do seem to be generally weak in this area.