Hi,
Thank you for your kind words!
1. I usually recommend a small custom code snippet, but this is a bit more complicated than that. It needs to be custom coded, but there is the plugin javascript API, that should help a bit with the proper event attachment.
2. This documentation will help you with that. In Elementor filter case the whole page needs to be loaded for the plugin to fetch the contents, so it very highly depends on the server response time, not as much on the overall search performance.
3. 3rd party widget may not work unfortuantely. The Elementor core posts widget uses certain hooks to give access the query, and if those hooks are not present (and they are usually not) in that other widget, then it will not work.