Insert wp gallery shortcode into custom textarea

Yes, it is possible :) I just needed to implement this functionality and come with your post.

Let's suppose do you have this HTML:

<input type="hidden" name="shortcode" value="{{ options.shortcode }}"/>
<br />
<a href="#" class="btn gallery-configure">
     Configure Gallery
</a>

The matching JS to trigger the Gallery edit modal and save the shortcode into the proper field would be:

/**
 * Gets initial gallery-edit images. Function modified from wp.media.gallery.edit
 * in wp-includes/js/media-editor.js.source.html
 *
 * @params {*} shortcode
 * @return {*}
 */
function select(shortcode) {
    var shortcode = wp.shortcode.next('gallery', shortcode);
    var defaultPostId = wp.media.gallery.defaults.id;
    var attachments;
    var selection;

    // Bail if we didn't match the shortcode or all of the content.
    if (!shortcode) {
        return;
    }

    shortcode = shortcode.shortcode;

    if (typeof shortcode.get('id') != 'undefined' && typeof defaultPostId != 'undefined') {
        shortcode.set('id', defaultPostId);
    }

    attachments = wp.media.gallery.attachments(shortcode);
    selection = new wp.media.model.Selection(attachments.models, {
        props   : attachments.props.toJSON(),
        multiple: true
    });

    selection.gallery = attachments.gallery;

    /*
     * Fetch the query's attachments, and then break ties from the query to allow for sorting.
     */
    selection.more().done(function () {
        // Break ties with the query.
        selection.props.set({
            query: false
        });
        selection.unmirror();
        selection.props.unset('orderby');
    });
    return selection;
}

var $body = jQuery('body');

// Media uploaders.
$body.on('click', '.gallery-configure', function () {
    var $this_button = jQuery(this);

    jQuery('.media-modal-close').trigger('click');

    var selection = var selection = select($this_button.parent().find('[name="shortcode"]').val()); // Get the saved selection here.
    var frame = wp.media({
        frame   : 'post',
        title   : wp.media.view.l10n.editGalleryTitle,
        multiple: false,
        state   : 'gallery-edit',
        editing : true,

        selection: selection
    });

    frame.on('update', function () {
        var controller = frame.states.get('gallery-edit');
        var library = controller.get('library');
        var new_shortcode = wp.media.gallery.shortcode(library).string(); // Get the new/updated shortcode here.

        $this_button.parent().find('[name="shortcode"]').val(new_shortcode);
    });

    frame.open();
    return false;
});

Hope this helps you, it took me some time to figure it out :)

I'm sure it's possible, but will require some work with both PHP and jQuery. This plugin by Thomas Griffin to insert an image into a meta box would be a very good starting point.

You'll need to do some digging around in the js files in wp-admin/js and in wp-admin/includes/media.php.