Reuse ChatComposer in request_user_input overlay (#9892)

Reuse the shared chat composer for notes and freeform answers in
request_user_input.

- Build the overlay composer with ChatComposerConfig::plain_text.
- Wire paste-burst flushing + menu surface sizing through the bottom
pane.

codex-rs/tui/src/bottom_pane/selection_popup_common.rs

@@ -42,6 +42,11 @@ pub(crate) fn menu_surface_inset(area: Rect) -> Rect {
     area.inset(Insets::vh(MENU_SURFACE_INSET_V, MENU_SURFACE_INSET_H))
 }
 
+/// Total vertical padding introduced by the menu surface treatment.
+pub(crate) const fn menu_surface_padding_height() -> u16 {
+    MENU_SURFACE_INSET_V * 2
+}
+
 /// Paint the shared menu background and return the inset content area.
 ///
 /// This keeps the surface treatment consistent across selection-style overlays