1.Create a single-(cpt).php in your theme, duplicate a your single.php and rename it as per your CPT name, so this page template will use to show your CPT queries.
For more details refer the wordpress codex.
https://codex.wordpress.org/Category_Templates
To create CPT arcive page follow the same proceduare.
archive-(cpt).php
For Taxonimies
taxonomy-(cpt).php
CPT is your custom post type name.
Check below article to get search results VIA ajax.
https://premium.wpmudev.org/blog/how-to-use-ajax-in-wordpress-to-load-search-results/?ptm=c&utm_expid=3606929-108.O6f5ypXuTg-XPCV9sY1yrw.2&utm_referrer=https%3A%2F%2Fwww.google.com%2F