Split ChatWidget state into focused modules (#21866)

## Summary

`ChatWidget` has been carrying several independent domains in one large
state bag: transcript bookkeeping, turn lifecycle, queued input, status
surfaces, connectors, review mode, and protocol dispatch. That makes
otherwise-local changes hard to reason about because unrelated fields
and side effects live beside each other in `chatwidget.rs`.

This is the first cleanup PR in a larger decomposition effort. It does
not try to make `chatwidget.rs` small in one sweep; instead, it
establishes focused state boundaries that later handler, popup,
rendering, and effect-synchronization extractions can build on.

This PR keeps `ChatWidget` as the composition layer while moving focused
state into smaller `codex-tui` modules. The widget still owns effects
that touch the bottom pane, app events, command submission, redraw
scheduling, and terminal-title updates.

## Changes

- Add focused state modules under `codex-rs/tui/src/chatwidget/` for
input queues, turn lifecycle, transcript bookkeeping, status state,
connectors, review mode, and app-server protocol dispatch.
- Update `ChatWidget` to hold grouped state structs and route
input/lifecycle/status operations through those focused helpers.
- Move app-server notification dispatch into `chatwidget/protocol.rs`
while leaving feature handlers and side effects on `ChatWidget`.
- Replace the large manual `ChatWidget` test literal with the normal
constructor plus narrow test overrides, so future state moves do not
require every field to be restated in test setup.
- Update existing tests to access the new grouped state or narrower
helpers without changing snapshot behavior.

## Longer-term direction

Follow-up PRs can continue shrinking `chatwidget.rs` by moving behavior,
not just state, into focused modules:

- Extract input/submission flow, turn/stream handling, and tool-cell
lifecycles into domain modules that call the new state reducers.
- Move popup/settings builders and rendering helpers out of the main
widget file so `ChatWidget` stays focused on composition.
- Reduce direct `BottomPane` mutation by applying domain-specific sync
outputs at clearer boundaries.

codex-rs/tui/src/chatwidget/tests.rs

@@ -149,7 +149,6 @@ pub(super) use codex_protocol::config_types::CollaborationMode;
 pub(super) use codex_protocol::config_types::ModeKind;
 pub(super) use codex_protocol::config_types::Personality;
 pub(super) use codex_protocol::config_types::ServiceTier;
-pub(super) use codex_protocol::config_types::Settings;
 pub(super) use codex_protocol::models::FileSystemPermissions;
 pub(super) use codex_protocol::models::MessagePhase;
 pub(super) use codex_protocol::models::NetworkPermissions;
@@ -179,7 +178,6 @@ pub(super) use serde_json::json;
 pub(super) use serial_test::serial;
 pub(super) use std::collections::BTreeMap;
 pub(super) use std::collections::HashMap;
-pub(super) use std::collections::HashSet;
 pub(super) use std::path::PathBuf;
 pub(super) use tempfile::NamedTempFile;
 pub(super) use tempfile::tempdir;