feat: add mobile application structure with emotion picker and API integration

- Added favicon and icon assets for the mobile app.
- Implemented EmotionPicker component for selecting emotions with child options.
- Created hooks for fetching emotions, feeds, and rapport data.
- Established API service for handling requests and authentication.
- Integrated Zustand for state management of user authentication.
- Defined color palette for the application.
- Set up TypeScript types for various entities including User, Emotion, and Feed.
- Configured Babel and TypeScript for the mobile project.
- Added package.json with necessary dependencies for Expo and React Native.

mobile/app/index.tsx

@@ -0,0 +1,7 @@
+import { Redirect } from 'expo-router';
+import { useAuthStore } from '@/lib/auth-store';
+
+export default function Index() {
+  const { isAuthenticated } = useAuthStore();
+  return <Redirect href={isAuthenticated ? '/(tabs)' : '/auth/login'} />;
+}