Implements issue #160 with a small testable SPA slice.
## Included
- Extend typed API client with write endpoints for join + start round
- Add vertical-slice SPA controller for lobby hydration, join, and round start
- Add loading + error state handling for join/start actions
- Add tests for happy path and failure states
## Checks
- cd frontend && npm test
- cd frontend && npm run build
Closes #160
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Implements issue #160 with a small testable SPA slice.
Included
Checks
Closes #160
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Automatisk review (REVIEW-bot): Grøn CI/checks og ingen blockers identificeret i denne gennemgang. Merge-klar.