feat(chat): auto-focus input on new session draft and enable double-click to switch tabs

[gsxdsm]

Summary

• Fix the session button first going back to a previous chat when in a worktree on mobile
• Auto-focus the chat textarea when a new session draft opens, with mobile friendly focus behavior to prevent viewport jump
• Enable double-click/double-tap on session items to switch to the chat tab (mobile & desktop parity)
• Small layout adjustments to render panels based on active view in main layouts
• Wire up double-click handlers in MobileSessionStatusBar and SessionSidebar to trigger tab switch

Why

• Improves usability by placing the user directly in the chat input when starting a new session draft
• Provides a quick, discoverable way to switch from session list to chat, aligning mobile interactions with desktop expectations
• Keeps the UI streamlined by showing only the active view while preserving the ability to drill down to sessions or chat

Testing

• Not run locally
• Open a new session draft and verify the chat textarea focuses automatically; on mobile, ensure focus does not scroll the viewport
• Double-click/double-tap a session in the status bar and in the session list to switch to the chat view
• Confirm layout renders correctly when toggling between sessions and chat views on both mobile and desktop
• Ensure existing keyboard navigation and accessibility attributes are preserved