Search form UX Improvemnts

[Raghaddahi]

Fixes #250

Description

This PR improves the search form by the following:

• Focus the search input automatically when opening the search modal
• Clear search input and results when the modal is closed
• Add a search icon inside the search bar
• Add a loading indicator during search requests
• Add animated reveal transitions for search results
• Move the results count to a separate line to prevent the text from getting over the number of results

Testing

Screencast.From.2026-05-11.18-07-11.mp4

AI usage

I used Opencode in helping with understanding the issues and files and in writing the code.