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/global-bottom-nav.tsx

@@ -3,6 +3,7 @@
 import { useEffect, useState } from "react";
 
 import { AppNav } from "@/components/layout/app-nav";
+import { AppDataPrefetcher } from "@/components/layout/app-data-prefetcher";
 import {
   NavOffsetProvider,
   useNavOffsetRef,
@@ -55,6 +56,7 @@ export function GlobalBottomNav() {
 
   return (
     <NavOffsetProvider enabled={showNav}>
+      <AppDataPrefetcher enabled={showNav} />
       {showNav ? <MeasuredAppNav /> : null}
     </NavOffsetProvider>
   );