Add doctor integration readiness summary

[Widthdom]

Summary

• Add cdidx doctor --integrations as a read-only integration readiness summary with human and typed JSON output for project/workspace freshness, Git hooks, MCP, LSP, watch, and extension/plugin state.
• Add bounded output, default path/secret redaction, --show-paths, --max-json-bytes, and CI-friendly --check behavior without starting integrations or plugin workers.
• Make process-free Git readiness checks conservative and accurate for HEAD changes, ancestor ignore rules, nested project roots, linked worktrees, sparse checkout/index, and unsupported split-index metadata.
• Return stable warning diagnostics for malformed workspace manifests and actionable hook executable failures.

Validation

• dotnet build CodeIndex.sln -c Release -p:UseSharedCompilation=false
• Focused Issue Add an integration readiness summary to cdidx doctor #5102 tests on .NET 8: 8 passed
• Focused Issue Add an integration readiness summary to cdidx doctor #5102 tests on .NET 9: 8 passed
• Full Release suite on .NET 9: 11,262 passed, 423 skipped, 0 failed
• Full Release suite on .NET 8: 11,740 passed, 7 skipped; one unrelated timing-sensitive fake-Git timeout test failed in the combined run and passed on isolated retry
• dotnet run --project tools/CodeIndex.Changelog -- check
• dotnet format CodeIndex.sln --no-restore --verify-no-changes
• git diff --check
• cdidx status --check --json
• cdidx workspace status --check --json
• Two bounded codex review --base origin/main adversarial-review rounds completed; all actionable findings were addressed and covered by tests

Documentation and changelog

• Updated the English and Japanese integration-doctor documentation in README.md, USER_GUIDE.md, and DEVELOPER_GUIDE.md, plus the testing contract in TESTING_GUIDE.md.
• Added changelog.d/unreleased/5102.added.md.

Follow-up candidates

• None.

Fixes #5102