[New docs] Site header: Implement basic design changes to match the custom theme

[tkajtoch]

Summary

Implement basic design changes to match the site header from the design file.

⚠ *Please note that this design file is not considered final, and there may be additional changes needed before releasing the new documentation site.

The mobile variant does not have an existing design. Please adjust the design on mobile devices to fit the screen.

Acceptance criteria

• The default docusaurus site header should be replaced with a custom one matching the design linked above
  • Version selector and search bar will be implemented separately and should be skipped for now
  • The cheer button icon should redirect to https://github.com/elastic/eui/tree/main/changelogs until the changelog page is implemented
• The mobile version should not overflow and should look consistent with the rest of the design