Custom Wordpress Loop not displaying when in Page Template

I don't know what you mean by "custom WordPress loop" but the code is running for what WordPress already queried for posts and accordingly on your custom page this is not done! So have_posts() returns false. To query "manually" for all posts you need to do the following:

$the_query = new WP_Query( $args );
if ( $the_query->have_posts() ) {

and the rest goes from <?php $count = 0; ?> on. for more info on WP_Query see the the following WP_Query

As pointed in the previous answers, you can use WP_Query to make a custom query for posts, custom post types (CPT) and pages:

$the_query = new WP_Query( array(
    'post_type'      => 'post',
    'posts_per_page' => 10,
    // other args here
) );

And use The Loop to display the posts:

if ( $the_query->have_posts() ) {
    while ( $the_query->have_posts() ) {
        $the_query->the_post();
        //
        // Post Content here
        //
    } // end while
} // end if

Now referring to this:

Partially Working WP Query Code

Here's my website where this code will appear but seems the not working

I think you meant to say, "the pagination is not working", right? Because it's not:

1. Because you're using the global $wp_query object with your pagination.

2. In your WP_Query construct, you didn't set the paged parameter which is needed for the pagination to work properly.

So here's how it should be:

$current_page = max( 1, get_query_var( 'paged' ) ); // the current page
$the_query = new WP_Query( array(
    'post_type'      => 'post',
    'posts_per_page' => 10,
    'paged'          => $current_page,
    // other args here
) );

And then use the $current_page with paginate_links() — you can also see here, I used $the_query and not $wp_query when retrieving/specifying the max number of pages:

echo paginate_links( array(
    'current'  => $current_page,
    'total'    => $the_query->max_num_pages, // here I don't use $wp_query
    // other args here
) );

And below is a working code you can use in place of your partially working code (the one in between the <!-- START of WP Query --> and <!-- END of WP Query -->):

<?php
$current_page = max( 1, get_query_var( 'paged' ) );
$the_query = new WP_Query( array(
    'post_type'      => 'post',
    'posts_per_page' => 10,
    'paged'          => $current_page,
    // other args here
) );

if ( $the_query->have_posts() ) :
    $count = 1;
    while ( $the_query->have_posts() ) : $the_query->the_post();
        if ( 1 === $count ) :
        ?>
        <div class="item item1" style="background: red; color: #fff;">
            <span>Post 1</span> <?php the_title( '<h3>', '</h3>' ); ?>
        </div>
        <?php
        elseif ( 2 === $count ) :
        ?>
        <div class="item item2" style="background: orange; color: #fff;">
            <span>Post 2</span> <?php the_title( '<h3>', '</h3>' ); ?>
        </div>
        <?php
        // other conditions here
        else :
        ?>
        <div class="item item3" style="background: yellow; color: #666;">
            <span>Post <?php echo $count; ?></span>
            <?php the_title( '<h3>', '</h3>' ); ?>
        </div>
        <?php
        endif;
        $count++;
    endwhile;
?>
<p>Pagination:</p>
<?php
    $big = 999999999; // need an unlikely integer
    echo paginate_links( array(
        'base'     => str_replace( $big, '%#%', esc_url( get_pagenum_link( $big ) ) ),
        'format'   => '?paged=%#%',
        'current'  => $current_page,
        'total'    => $the_query->max_num_pages,
        'type'     => 'list',
        'end_size' => 3,
    ) );
?>
<?php else : ?>
    <p><?php esc_html_e( 'Sorry, no posts matched your criteria.' ); ?></p>
<?php endif;

wp_reset_postdata();
?>

You need to define the $args. You can find a list of items you can use in WP_Query here https://www.billerickson.net/code/wp_query-arguments/

See below:

// WP_Query arguments
$args = array(
    'post_type'              => array( 'post' ),
);

// The Query
$query = new WP_Query( $args );

// The Loop
if ( $query->have_posts() ) {
    while ( $query->have_posts() ) {
        $query->the_post();
        // do something
    }
} else {
    // no posts found
}

// Restore original Post Data
wp_reset_postdata();