Do we have to register JQuery if we want to use it in wp_enqueue_script()?

You don’t need to register it but in your own js file, replace the first $ to jQuery.

jQuery is loaded in No Conflict Mode

jQuery(document).ready(function($) {
    // Your code here. Use $ as normal.
});

And if you use anonymous functions, do it like this

(function($) {
   // Your code here. Use $ as normal.
})(jQuery);

Also, you need to use relative paths.