Finish interrupted features: tab prefetch, pet viewport, and home logo.

Add shared quiz/habits tab cache with nav hover prefetch and AppDataPrefetcher,
make the pet screen fill and center the viewport, and point the HaBit logo to /.

components/layout/app-data-prefetcher.tsx

@@ -0,0 +1,36 @@
+"use client";
+
+import { useEffect } from "react";
+
+import { HABIT_PET_DATA_UPDATED_EVENT } from "@/lib/app-events";
+import {
+  invalidateAppTabDataCache,
+  prefetchAppTabData,
+} from "@/lib/app-tab-data-cache";
+
+type AppDataPrefetcherProps = {
+  enabled: boolean;
+};
+
+export function AppDataPrefetcher({ enabled }: AppDataPrefetcherProps) {
+  useEffect(() => {
+    if (!enabled) {
+      return;
+    }
+
+    void prefetchAppTabData();
+
+    const handleDataUpdated = () => {
+      invalidateAppTabDataCache();
+      void prefetchAppTabData({ force: true });
+    };
+
+    window.addEventListener(HABIT_PET_DATA_UPDATED_EVENT, handleDataUpdated);
+
+    return () => {
+      window.removeEventListener(HABIT_PET_DATA_UPDATED_EVENT, handleDataUpdated);
+    };
+  }, [enabled]);
+
+  return null;
+}