Deploy push relay: dev-shell tooling, real relay URL, /health route

- flake.nix: add nodejs_22, firebase-tools, and google-cloud-sdk to the dev
  shell so the relay can be tested, built, deployed, and administered locally
- backend settings: point default_relay_url at the deployed relay
- rename the relay liveness route /healthz -> /health: Google Front End
  reserves /healthz and returns its own 404 before the request reaches Cloud
  Run, so the probe was unreachable (verified live; /v1/push and the FCM path
  work end-to-end)

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>

flake.nix

@@ -69,6 +69,8 @@
           androidSdk
           jdk17
           nodejs_22 # push relay (push_relay/): runs npm install/test/build; bundles npm
+          firebase-tools # push relay: firebase CLI for emulator (serve) and deploy
+          google-cloud-sdk # push relay: gcloud for Cloud Run/IAM admin (e.g. invoker)
           claude-code
           clippy # Rust linter
           pythonEnv