Show timer remaining per block and fix single-click block switching

- Block list on both dashboards now shows time remaining on each block's
  timer (allocated duration minus elapsed) instead of total duration;
  the active block counts down live every second
- Fix block switching requiring 2 clicks: replace separate pause+start
  requests with a single start request; backend implicitly records a
  pause event for the previous block atomically
- Export blockElapsedCache from store so views can compute per-block
  elapsed for both running and paused blocks

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

frontend/src/views/TVView.vue

@@ -86,6 +86,7 @@
             :block="block"
             :is-current="block.id === scheduleStore.session?.current_block_id"
             :is-completed="scheduleStore.completedBlockIds.includes(block.id)"
+            :elapsed-seconds="blockElapsed(block)"
             compact
           />
         </div>
@@ -179,6 +180,14 @@ const currentSubjectOptions = computed(() =>
   scheduleStore.currentBlock?.subject?.options || []
 )
 
+function blockElapsed(block) {
+  const currentId = scheduleStore.session?.current_block_id
+  if (block.id === currentId && scheduleStore.blockStartedAt && !scheduleStore.isPaused) {
+    return scheduleStore.blockElapsedOffset + Math.floor((now.value - scheduleStore.blockStartedAt) / 1000)
+  }
+  return scheduleStore.blockElapsedCache[block.id] || 0
+}
+
 // WebSocket
 const { connected: wsConnected } = useWebSocket(childId, (msg) => {
   scheduleStore.applyWsEvent(msg)