Add a Docker image build-and-push CI step

[james2177]

Description

A working Dockerfile exists at the repo root, but .github/workflows/ci.yml never builds it — there's no CI signal if a code change breaks the Docker build, and no automated image publishing for deployment.

Requirements and context

• Add a build step at minimum (fail CI on broken Dockerfile); pushing to a registry can be a separate follow-up decision requiring registry credentials

Suggested execution

1. Fork the repo and create a branch

   git checkout -b ci/docker-build-step
   

2. Implement changes
   Relevant files:

• .github/workflows/ci.yml

3. Test and commit

   • Run npm run lint && npm run typecheck && npm test && npm run test:e2e
   • Include test output / screenshots or logs demonstrating the fix in the PR

   Example commit message:

   ci: add Docker image build verification to CI
   

Guidelines

• Assignment required before starting — comment to claim this issue
• PR description must include: Closes #[this issue's number]
• Timeframe: 96 hours from assignment
• Complexity: High (200 points)

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Part of the Drips Wave for vortex-backend.