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/common/mcp_process.rs

@@ -174,7 +174,7 @@ impl McpProcess {
             client_info,
             Some(InitializeCapabilities {
                 experimental_api: true,
-                opt_out_notification_methods: None,
+                ..Default::default()
             }),
         )
         .await