Fix timer reset on refresh and sync between dashboard and TV view

- Backend computes block_elapsed_seconds server-side from timer_events
- Store tracks blockStartedAt (ms) + blockElapsedOffset (seconds) instead
  of a client-side counter; updated correctly on start/pause/resume/end
- TimerDisplay derives elapsed from store props so both views always agree
- Add compact timer display to dashboard session card
- Add isPaused/pause-resume logic to dashboard Pause/Resume buttons

Co-Authored-By: Claude Sonnet 4.6 <noreply@anthropic.com>

backend/app/routers/dashboard.py

@@ -2,7 +2,7 @@
 Public dashboard endpoint — no authentication required.
 Used by the TV view to get the initial session snapshot before WebSocket connects.
 """
-from datetime import date
+from datetime import date, datetime
 
 from fastapi import APIRouter, Depends, HTTPException
 from sqlalchemy.ext.asyncio import AsyncSession
@@ -40,6 +40,7 @@ async def get_dashboard(child_id: int, db: AsyncSession = Depends(get_db)):
 
     blocks = []
     completed_ids = []
+    block_elapsed_seconds = 0
 
     if session and session.template_id:
         blocks_result = await db.execute(
@@ -57,9 +58,34 @@ async def get_dashboard(child_id: int, db: AsyncSession = Depends(get_db)):
         )
         completed_ids = [e.block_id for e in events_result.scalars().all() if e.block_id]
 
+    # Compute elapsed seconds for the current block from timer_events
+    if session and session.current_block_id:
+        tick_result = await db.execute(
+            select(TimerEvent)
+            .where(
+                TimerEvent.session_id == session.id,
+                TimerEvent.block_id == session.current_block_id,
+                TimerEvent.event_type.in_(["start", "resume", "pause"]),
+            )
+            .order_by(TimerEvent.occurred_at)
+        )
+        tick_events = tick_result.scalars().all()
+        last_start = None
+        elapsed = 0.0
+        for e in tick_events:
+            if e.event_type in ("start", "resume"):
+                last_start = e.occurred_at
+            elif e.event_type == "pause" and last_start:
+                elapsed += (e.occurred_at - last_start).total_seconds()
+                last_start = None
+        if last_start:
+            elapsed += (datetime.utcnow() - last_start).total_seconds()
+        block_elapsed_seconds = int(elapsed)
+
     return DashboardSnapshot(
         session=session,
         child=child,
         blocks=blocks,
         completed_block_ids=completed_ids,
+        block_elapsed_seconds=block_elapsed_seconds,
     )