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.

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.