Fix MCP request lifecycle races

[Widthdom]

Summary

• Make MCP running state visible across threads and serialize stdout frame writes.
• Add bounded MCP request timeouts, including batch items, while isolating timed transport requests from the shared DB connection.
• Drain stdio EOF in-flight work with bounded post-cancel settling and add a bilingual changelog fragment.

Validation

• dotnet build CodeIndex.sln -c Release -p:UseSharedCompilation=false
• dotnet test tests/CodeIndex.Tests/CodeIndex.Tests.csproj --filter "FullyQualifiedName~McpServerTests"
• dotnet test CodeIndex.sln -c Release -p:UseSharedCompilation=false
• dotnet run --project tools/CodeIndex.Changelog -- check
• codex adversarial review: No blocking/actionable issues found.

Notes

• No AGENT.md, CLAUDE.md, or AGENT_GUIDE.md changes.
• Local index refresh hit E012_INTERRUPTED; existing open issue Index refresh can hang on large extraction phase #2665 covers that follow-up, so no duplicate issue was opened.

Fixes #1736
Fixes #1737
Fixes #1738
Fixes #1742