Hi Raj,
So the first option needs to be coded via Elementor. All the plugin does is passes the arguments to the elementor widget to get the items, and then Elementor gives back the output, which the plugin puts back to the page. The output is generated by Elementor, therefore it needs to be coded on that end. I am unfortunately not aware of any hooks on how to modify that.
The second option should be doable. You can use the search results shortcode to place the result somewhere on the same page. You can simply replace the relementor widget with that. If you enable auto populate feature, results will be displayed initially on page load as well. On that layout I recommend turning off this option, so the results box does not close.