To make something similar, you can use the search filters. There are a few example on the demo with tutorials, like this and this, which you can follow.
The results page is fully controlled by your theme. Check your theme options on what changes it may offer for that page. Some themes may even have page builder features to modify the output of the results archive.
q1) You can add a search button as a filter element. You can also add a search text next to the magnifier.
q2) The plugin redirects ot the default results page, so if you change the results page via the Elementor, then you don’t have to do anything, the plugin automatically redirects to the results page.