fix: improve home header responsive layout

[kunal-9090]

Linked issue

Closes #1228

GSSoC

• Source: official GSSoC issues page
• Labels: gssoc, level:intermediate, type:design, type:refactor

Summary

• Updates the home header to use a lighter, higher-contrast visual treatment with a subtle bottom border.
• Adds compact-width handling so the search input, filter control, and notification action do not squeeze or overlap on narrow screens.
• Reduces icon and input sizing on compact devices while keeping the notification action from shrinking.
• Removes an unused header import so the changed file passes ESLint cleanly.

Validation

• pnpm install --ignore-scripts
• pnpm exec eslint src/components/HomeScreenHeader.tsx
• Attempted pnpm run type-check; it is blocked by pre-existing unrelated syntax errors in src/hooks/useSendOtp.ts, src/screens/article/ArticleScreen.tsx, and src/screens/auth/NewPasswordScreen.tsx.
• Attempted pnpm run lint; Expo lint exits successfully but prints an environment warning because yarnpkg is unavailable in this local shell.

[github-actions]

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