How to loop through custom post type items and add them to a slider if they exist

Sure thing, couple of pointers:

1. No need for an empty else – just go straight to the endif
2. == true is a redundant condition. if ( $var ) and if ( $var == true ) are the same. Note that $var === true is different though (read up on comparison operators)
3. Use site_url( $path ) instead of get_site_url() . $path – this will ensure the URL is properly concatenated (if $path had no preceding slash you’d end up with example.commypath)
4. Sidenote: might want to re-think your mobile images technique – regardless of your CSS/JS, any device is going to request both images

And save yourself even more typing by assigning values within the condition. For example:

// This...
if ( get_post_meta( $id, $field ) )
    echo get_post_meta( $id, $field );

// Less function overhead, neater
$value = get_post_meta( $id, $field );
if ( $value )
    echo $value;

// Even neater
if ( $value = get_post_meta( $id, $field ) )
    echo $value;

The result:

<?php

for ( $i = 1; $i <= 10; $i++ ) {
    if ( $image = get_post_meta( $post->ID, "{$i}-carousel-image", true ) ) : ?>

        <div class="item">
            <?php if ( $title = get_post_meta( $post->ID, "{$i}-carousel-title", true ) ) : ?>
                <div class="display-block">
                    <div class="text-block slider">
                        <h3 class="slider-title"><?php echo $title ?></h3>
                        <p><?php echo get_post_meta( $post->ID, "{$i}-carousel-text", true ) ?></p>
                        <?php if ( $text = get_post_meta( $post->ID, "{$i}-carousel-button-text", true ) ) : ?>
                            <a href="https://wordpress.stackexchange.com/questions/237132/<?php echo esc_url( site_url( get_post_meta( $post->ID,"{$i}-carousel-button-link", true ) ) ) ?>" class="button primary"><?php echo $text ?></a>
                        <?php endif ?>
                    </div>
                </div>

            <?php endif ?>
            <img class="mobile-hide" src="<?php echo esc_url( $image ) ?>" alt="<?php echo esc_attr( get_post_meta( $post->ID, "{$i}-carousel-image-alt", true ) ) ?>" />
            <img class="mobile-show" src="<?php echo esc_url( get_post_meta( $post->ID, "{$i}-carousel-image-mobile", true ) ) ?>" alt="<?php echo esc_attr( get_post_meta( $post->ID, "{$i}-carousel-image-alt", true ) ) ?>" />
        </div>

    <?php
    endif;
}