Add sidebar tabs and nonblocking side-chat creation (#59767)

## Why

Starting a side chat waits for fork and setup requests. Keep the workspace responsive during those requests and let users return to an existing side chat through the sidebar chooser.

## What changed

- Add a clickable `Side chat` tab and `+` chooser, plus the configurable `focus_sidebar_tabs` action bound to `leader t` by default.
- Create workspace side chats in the background, showing loading and failure states with `Enter` to retry and `Esc` to go back. Preserve opening prompts and avoid taking focus when the user has moved away.
- Retain inherited permission profiles and defer startup notifications until the side chat is registered on its owning backend.
- Quiet the parent composer while the chooser has focus and prevent completion popups or repeated activation keys from consuming chooser input.

## Testing

Extend tests and snapshots for loading and failure states, returning to an existing chat, draft preservation, inherited permissions, focus transitions, and deferred MCP startup notifications.

GitOrigin-RevId: 244bcd3f170819e2260f21dc1041cf8899fe048c

codex-rs/core/config.schema.json

@@ -5327,6 +5327,7 @@
               "focus_pane_left": null,
               "focus_pane_right": null,
               "focus_sidebar_content": null,
+              "focus_sidebar_tabs": null,
               "hide_pane": null,
               "leader": null,
               "maximize_pane_width": null,
@@ -6211,6 +6212,14 @@
           ],
           "description": "Focus the side conversation, or hide it when already focused."
         },
+        "focus_sidebar_tabs": {
+          "allOf": [
+            {
+              "$ref": "#/definitions/KeybindingsSpec"
+            }
+          ],
+          "description": "Open the sidebar tab chooser."
+        },
         "hide_pane": {
           "allOf": [
             {
@@ -6485,6 +6494,7 @@
             "focus_pane_left": null,
             "focus_pane_right": null,
             "focus_sidebar_content": null,
+            "focus_sidebar_tabs": null,
             "hide_pane": null,
             "leader": null,
             "maximize_pane_width": null,