Editing the custom background CSS

Yes, that is possible. Take a look at the codex and you will see that you can pass arguments with add_theme_support( 'custom-background'). One of them is the callback function that generates the <style> tags: _custom_background_cb. You can pass your own function as an argument to add_theme_support.

Here is the code of the original function (retrieved from WP 4.5, please check if you read this later):

function _custom_background_cb() {
    // $background is the saved custom image, or the default image.
    $background = set_url_scheme( get_background_image() );

    // $color is the saved custom color.
    // A default has to be specified in style.css. It will not be printed here.
    $color = get_background_color();

    if ( $color === get_theme_support( 'custom-background', 'default-color' ) ) {
        $color = false;
    }

    if ( ! $background && ! $color )
        return;

    $style = $color ? "background-color: #$color;" : '';

    if ( $background ) {
        $image = " background-image: url('$background');";

        $repeat = get_theme_mod( 'background_repeat', get_theme_support( 'custom-background', 'default-repeat' ) );
        if ( ! in_array( $repeat, array( 'no-repeat', 'repeat-x', 'repeat-y', 'repeat' ) ) )
            $repeat="repeat";
        $repeat = " background-repeat: $repeat;";

        $position = get_theme_mod( 'background_position_x', get_theme_support( 'custom-background', 'default-position-x' ) );
        if ( ! in_array( $position, array( 'center', 'right', 'left' ) ) )
            $position = 'left';
        $position = " background-position: top $position;";

        $attachment = get_theme_mod( 'background_attachment', get_theme_support( 'custom-background', 'default-attachment' ) );
        if ( ! in_array( $attachment, array( 'fixed', 'scroll' ) ) )
            $attachment="scroll";
        $attachment = " background-attachment: $attachment;";

        $style .= $image . $repeat . $position . $attachment;
    }
?>
<style type="text/css" id="custom-background-css">
body.custom-background { <?php echo trim( $style ); ?> }
</style>
<?php
}

Copy this, rename it wpse228339_callback_function (or so), change the last lines where the css is printed and pass it as a callback like this:

$defaults = array(
    'default-color'          => '',
    'default-image'          => '',
    'default-repeat'         => '',
    'default-position-x'     => '',
    'default-attachment'     => '',
    'wp-head-callback'       => 'wpse228339_callback_function',
    'admin-head-callback'    => '',
    'admin-preview-callback' => ''
);
add_theme_support( 'custom-background', $defaults );