Healthchecks are failing, even though the service is running and open via fw rules

Healthchecks actually originate from GCP directly, rather than the load balancer instance itself.  So these networks must be whitelisted in the firewall rules:

• 35.191.0.0/16
• 130.211.0.0/22

The LB works in the same region, but does not respond from different regions

By default, load balancers operate in regional-only mode.  To switch to global, edit the frontend properties and look for this radio button: