Some HTTP proxies can do retries -- if a connection to one backend fails, it is retried on a different backend. Can PgDog (or PgBouncer, or any other tool) do something similar -- if there's a "database server shutting down" error or a connection reset, retry it on another backend?

Not currently, but we can add this. One thing we have to be careful of is to not retry requests that are executing inside transactions, but otherwise this would be a great feature.