Background agents that tell the truth, and a "needs input" dot that actually clears

Fixed

• Answering a follow-up question left the amber needs input dot on — the same run resumed in place, so nothing re-derived the flag.
• Switching away from a session that was waiting on an approval or a question cleared the dot; switching away from one that was not waiting could not clear a stale one. Both are now derived from the transcript.
• The background-agents strip rendered truncated session ids over "Waiting…" for chats it could not name, and its header said "running" even when every agent needed you.
• Files-changed counted in-flight and failed writes. Only finished, successful edits count, and Stop is hidden once the run has actually ended.
• Switching chats mid-flight dropped the in-flight spawn_agent card: the lanes kept running, the host card did not. Live tool previews are parked and restored with the session.
• A chat save with no owning workspace was dropped silently, so the transcript vanished from the sidebar on the next load.
• Clicking an open menu's trigger closed it on mousedown and reopened it on click.
• Opening a modal focused the first control, then jumped to the intended one.
• Arrow keys in a menu treated stray buttons (search-clear, header actions) as menu items.
• Two Browser panes shared one address-bar id, so labels and datalists collided.

New

• The background-agents strip uses the shared live-status line, sorts by urgency, persists collapse, and says Open / Stop instead of Focus. Parallel-lane runs stay in the Agents rail, not this strip.
• Menus, dialogs, and hover cards keep a single focus target on open, restore it on close, and let one Escape dismiss only the topmost surface.