feat(openapi): generate OpenAPI specs from TypeSpec sources

Migrate hand-written OpenAPI YAML to TypeSpec (.tsp) source files.
TypeSpec compiles to OpenAPI 3.0 YAML, enabling type-safe API definitions.

- Add typespec/ directory with api-server and fortee definitions
- Integrate TypeSpec build into `just gen` and `just build` pipelines
- Update backend handler code to match new generated type names
  (inlined error responses, separate GameType/ProblemLanguage enums)
- Regenerate frontend TypeScript types from new OpenAPI output

Co-Authored-By: Claude Opus 4.6 <noreply@anthropic.com>

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