Widgets won’t save

Question

This line of code is incorrect

<?php if ( !function_exists('redmenu_widget') || !dynamic_sidebar('redmenu_widget') ) : ?>

More to the point, your negative operator (!) is wrong here

That line of code simply translates to

If the function redmenu_widget does not exist OR if the sidebar redmenu_widget does not exists, do something

You should just drop the operator and also the !function_exists('redmenu_widget') part. You should also rather do a check to see if a sidebar is active. That part of code should look like this

<aside id="widgets"> 
    <?php if ( is_active_sidebar('redmenu_widget') ) : ?>
        <div id="primary" class="widget-area">
            <ul class="xoxo">
                <?php dynamic_sidebar('redmenu_widget'); ?>
            </ul>
        </div><!-- #primary .widget-area -->
        <?php endif; ?>   
</aside>

Also, your widget is a complete mess unfortunately. There are are lot of errors. You should check the Widget API and how to correctly create a widget. Also, when registering your widget, you want to use the class name.

EDIT

From your comments, you need a custom text widget. To accomplish this, simply copy the default text widget and customize as needed. Here is the default text widget, I have already renamed it, so you can just copy and paste it into your theme

class Red_Menu_Widget extends WP_Widget { 

    public function __construct() { 
        $widget_ops = array( 'classname' => 'redmenu_widget', 'description' =>  __('Rood menu rechts', 'redmenu_widget') ); 
        $control_ops = array( 'width' => 400, 'height' => 350 ); 
        parent::__construct( 'redmenu_widget', __('Rood menu rechts', 'redmenu_widget'),  $widget_ops, $control_ops ); 
    } 

    public function widget( $args, $instance ) { 

        /** This filter is documented in wp-includes/default-widgets.php */ 
        $title = apply_filters( 'widget_title', empty( $instance['title'] ) ? '' : $instance['title'], $instance, $this->id_base ); 

        /** 
         * Filter the content of the Text widget. 
         * 
         * @since 2.3.0 
         * 
         * @param string    $widget_text The widget content. 
         * @param WP_Widget $instance    WP_Widget instance. 
         */ 
        $text = apply_filters( 'widget_text', empty( $instance['text'] ) ? '' : $instance['text'], $instance ); 
        echo $args['before_widget']; 
        if ( ! empty( $title ) ) { 
            echo $args['before_title'] . $title . $args['after_title']; 
        } ?> 
            <div class="textwidget"><?php echo !empty( $instance['filter'] ) ? wpautop( $text ) : $text; ?></div> 
        <?php 
        echo $args['after_widget']; 
    } 

    public function update( $new_instance, $old_instance ) { 
        $instance = $old_instance; 
        $instance['title'] = strip_tags($new_instance['title']); 
        if ( current_user_can('unfiltered_html') ) 
            $instance['text'] =  $new_instance['text']; 
        else 
            $instance['text'] = stripslashes( wp_filter_post_kses( addslashes($new_instance['text']) ) ); // wp_filter_post_kses() expects slashed 
        $instance['filter'] = isset($new_instance['filter']); 
        return $instance; 

    public function form( $instance ) { 
        $instance = wp_parse_args( (array) $instance, array( 'title' => '', 'text' => '' ) ); 
        $title = strip_tags($instance['title']); 
        $text = esc_textarea($instance['text']); 
 ?> 
        <p><label for="<?php echo $this->get_field_id('title'); ?>"><?php _e('Title:'); ?></label> 
        <input class="widefat" id="<?php echo $this->get_field_id('title'); ?>" name="<?php echo $this->get_field_name('title'); ?>" type="text" value="<?php echo esc_attr($title); ?>" /></p> 

        <textarea class="widefat" rows="16" cols="20" id="<?php echo $this->get_field_id('text'); ?>" name="<?php echo $this->get_field_name('text'); ?>"><?php echo $text; ?></textarea> 

        <p><input id="<?php echo $this->get_field_id('filter'); ?>" name="<?php echo $this->get_field_name('filter'); ?>" type="checkbox" <?php checked(isset($instance['filter']) ? $instance['filter'] : 0); ?> />&nbsp;<label for="<?php echo $this->get_field_id('filter'); ?>"><?php _e('Automatically add paragraphs'); ?></label></p> 
 <?php 
    } 
 }

And then to register it

add_action( 'widgets_init', 'prefix_register_widgets' );
function prefix_register_widgets() {
    register_widget( 'Red_Menu_Widget' );
}

You can register your sidebar as per example in your question (just change the id as it might create conflict with your widget, call it something like redmenu-sidebar) and then call your sidebar as follows in your template file

<aside id="widgets"> 
    <?php if ( is_active_sidebar( 'redmenu_sidebar' ) ) : ?>
        <div id="primary" class="widget-area">
            <ul class="xoxo">
                <?php dynamic_sidebar('redmenu_sidebar'); ?>
            </ul>
        </div><!-- #primary .widget-area -->
        <?php endif; ?>   
</aside>

This should do it for you. Modify the code and widget to suit your needs

EDIT

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