Reapply "Add app-server transport layer with websocket support" (#11370)

Reapply "Add app-server transport layer with websocket support" with
additional fixes from https://github.com/openai/codex/pull/11313/changes
to avoid deadlocking.

This reverts commit 47356ff83c.

## Summary

To avoid deadlocking when queues are full, we maintain separate tokio
tasks dedicated to incoming vs outgoing event handling
- split the app-server main loop into two tasks in
`run_main_with_transport`
   - inbound handling (`transport_event_rx`)
   - outbound handling (`outgoing_rx` + `thread_created_rx`)
- separate incoming and outgoing websocket tasks

## Validation

Integration tests, testing thoroughly e2e in codex app w/ >10 concurrent
requests

<img width="1365" height="979" alt="Screenshot 2026-02-10 at 2 54 22 PM"
src="https://github.com/user-attachments/assets/47ca2c13-f322-4e5c-bedd-25859cbdc45f"
/>

---------

Co-authored-by: jif-oai <jif@openai.com>

codex-rs/app-server/tests/suite/v2/review.rs

@@ -5,8 +5,6 @@ use app_test_support::create_mock_responses_server_repeating_assistant;
 use app_test_support::create_mock_responses_server_sequence;
 use app_test_support::create_shell_command_sse_response;
 use app_test_support::to_response;
-use codex_app_server_protocol::CommandExecutionApprovalDecision;
-use codex_app_server_protocol::CommandExecutionRequestApprovalResponse;
 use codex_app_server_protocol::ItemCompletedNotification;
 use codex_app_server_protocol::ItemStartedNotification;
 use codex_app_server_protocol::JSONRPCError;
@@ -211,9 +209,7 @@ async fn review_start_exec_approval_item_id_matches_command_execution_item() ->
 
     mcp.send_response(
         request_id,
-        serde_json::to_value(CommandExecutionRequestApprovalResponse {
-            decision: CommandExecutionApprovalDecision::Accept,
-        })?,
+        serde_json::json!({ "decision": codex_core::protocol::ReviewDecision::Approved }),
     )
     .await?;
     timeout(