Depends on how they are configured, I think in some regions (where power outages are very rare), they are wired to sync up with external network, and without external network they shut down as well.

My understanding is that home batteries are not UPSes, they don't go through the battery. They have a switch between power company, solar, or battery. I think that means would be exposed to surge from power company.

You can install a whole house surge protector. Those go in the panel and would protect from different sources.

Yes. The power walls are like cheap UPS topology. You could still get whacked with a transient from the grid before the ATS decides to island the house.