feat(app-server): propagate traces across tasks and core ops (#14387)

## Summary

This PR keeps app-server RPC request trace context alive for the full
lifetime of the work that request kicks off (e.g. for `thread/start`,
this is `app-server rpc handler -> tokio background task -> core op
submissions`). Previously we lose trace lineage once the request handler
returns or hands work off to background tasks.

This approach is especially relevant for `thread/start` and other RPC
handlers that run in a non-blocking way. In the near future we'll most
likely want to make all app-server handlers run in a non-blocking way by
default, and only queue operations that must operate in order (e.g.
thread RPCs per thread?), so we want to make sure tracing in app-server
just generally works.

Depends on https://github.com/openai/codex/pull/14300

**Before**
<img width="155" height="207" alt="image"
src="https://github.com/user-attachments/assets/c9487459-36f1-436c-beb7-fafeb40737af"
/>


**After**
<img width="299" height="337" alt="image"
src="https://github.com/user-attachments/assets/727392b2-d072-4427-9dc4-0502d8652dea"
/>

## What changed

- Keep request-scoped trace context around until we send the final
response or error, or the connection closes.
- Thread that trace context through detached `thread/start` work so
background startup stays attached to the originating request.
- Pass request trace context through to downstream core operations,
including:
  - thread creation
  - resume/fork flows
  - turn submission
  - review
  - interrupt
  - realtime conversation operations
- Add tracing tests that verify:
  - remote W3C trace context is preserved for `thread/start`
  - remote W3C trace context is preserved for `turn/start`
  - downstream core spans stay under the originating request span
  - request-scoped tracing state is cleaned up correctly
- Clean up shutdown behavior so detached background tasks and spawned
threads are drained before process exit.

codex-rs/core/src/codex_thread.rs

@@ -19,6 +19,7 @@ use codex_protocol::protocol::AskForApproval;
 use codex_protocol::protocol::SandboxPolicy;
 use codex_protocol::protocol::SessionSource;
 use codex_protocol::protocol::TokenUsage;
+use codex_protocol::protocol::W3cTraceContext;
 use codex_protocol::user_input::UserInput;
 use std::path::PathBuf;
 use tokio::sync::Mutex;
@@ -67,6 +68,18 @@ impl CodexThread {
         self.codex.submit(op).await
     }
 
+    pub async fn shutdown_and_wait(&self) -> CodexResult<()> {
+        self.codex.shutdown_and_wait().await
+    }
+
+    pub async fn submit_with_trace(
+        &self,
+        op: Op,
+        trace: Option<W3cTraceContext>,
+    ) -> CodexResult<String> {
+        self.codex.submit_with_trace(op, trace).await
+    }
+
     pub async fn steer_input(
         &self,
         input: Vec<UserInput>,