refactor: stack mode indicator above room info

Creates cleaner vertical hierarchy in the navigation center section.

**Before:**
[Title▼]  [Battle] [Room]  [Players]

**After:**
[Title▼]  [Battle]  [Players]
          [Room]

Changes:
- Mode and Room now in flexDirection: 'column' with 4px gap
- Aligned to flex-end to keep right-aligned
- Creates more vertical breathing room
- Clearer visual grouping of related info

Result: Better visual hierarchy with mode indicator prominently on top
and room details nested below.

🤖 Generated with [Claude Code](https://claude.com/claude-code)

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

apps/web/src/components/nav/GameContextNav.tsx

@@ -165,13 +165,14 @@ export function GameContextNav({
           showMenu={!canModifyPlayers}
         />
 
-        {/* Mode + Room */}
+        {/* Mode + Room stacked vertically */}
         <div
           style={{
             display: 'flex',
-            alignItems: 'center',
-            gap: '8px',
+            flexDirection: 'column',
+            gap: '4px',
             marginLeft: 'auto',
+            alignItems: 'flex-end',
           }}
         >
           <GameModeIndicator gameMode={gameMode} shouldEmphasize={shouldEmphasize} showFullscreenSelection={false} />