Does a wordpress custom plugin is fired by default on php theme ajax request?

This will never work:

add_action( 'admin_init', 'my_ajax_checker', 10, 2);

function my_ajax_checker() {
    if( wp_doing_ajax() ) {
         add_action('plugins_loaded', 'atv_convert_multilingual');
    }
}

This is because by the time admin_init runs, plugins_loaded has already happened! Making this code would would require the code to travel backwards in time.

Instead it would be much simpler to do this:

add_action('plugins_loaded', 'atv_convert_multilingual');

Or this:

if( wp_doing_ajax() ) {
     add_action('plugins_loaded', 'atv_convert_multilingual');
}

Take a look at https://www.rarst.net/wordpress/wordpress-core-load/ for more information on the order that hooks fire in WordPress