feat(webui): improve chat page UX and message navigation

**Background**

This is a tracking issue for a set of UX improvements to the chat page, covering panel auto-open behavior, scroll-to-bottom navigation, session entry scroll positioning, text input layout optimization, and paginated message loading. These changes collectively improve the usability of the chat interface for sessions with long conversation histories and active task management.

**Changes**

- #2095 Auto-open the Task Panel when task updates arrive via CustomEvent
- #2096 Add a floating scroll-to-bottom button in the message list
- #2097 Auto-scroll to bottom when navigating to or opening a session
- #2098 Increase TextArea max visible rows and optimize Shift+Enter hint placement
- #2099 Support loading older messages via scroll-to-top when has_more is true

This is an umbrella issue, so start by reading linked items #2095, #2096, #2097, #2098, and #2099 for exact behavior. Then locate the chat web UI entry points that control the task panel, message list, session opening, textarea layout, and paged message loading (where has_more is handled). Run the web UI test/build flow after changes and confirm each listed UX behavior is implemented end to end.

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