Add /healthz healthcheck endpoint to FastAPI backend for container orchestration monitoring

[chigwell]

User Story
As a DevOps engineer maintaining containerized services,
I want to implement a healthcheck endpoint in the backend API
so that container orchestration tools can monitor service availability and automate recovery.

Background
The current FastAPI implementation in backend/app/main.py lacks a dedicated health probe endpoint. This prevents Kubernetes/Docker from performing liveness/readiness checks, risking undetected service degradation during deployments or runtime failures. The existing GET / endpoint (line 14) only serves basic functionality without operational awareness capabilities. Missing healthchecks could lead to prolonged downtime if containers remain scheduled while the application is unresponsive.

Acceptance Criteria

• Add /healthz endpoint to backend/app/main.py that returns HTTP 200 + JSON {"status": "healthy"}
• Endpoint must execute within 2 seconds for Kubernetes probe timeouts
• Update backend Dockerfile to expose healthcheck configuration if required
• Verify endpoint accessibility via curl -I http://localhost:80/healthz
• Confirm Kubernetes would interpret HTTP 200 as healthy status through manual testing
• Ensure endpoint doesn't require authentication and has minimal dependencies
• Add endpoint documentation to the API section of README.md
• Maintain existing CORS configuration (lines 8-13) for the new endpoint
• Test container restart behavior when intentionally breaking the healthcheck