<?php
function rm_category_posts(){
   $rm_args = array(
        'cat'               => ID,
        'posts_per_page'    => 5
    );
    query_posts( $rm_args );
    if( have_posts() ): 
        $rm_counter = 0;
        ?><h2 class="rm_category_name"><?php echo $rm_category->name;?></h2>
        <ul id="rm_category_posts"><?php
        while ( have_posts() ) : the_post();
            if ( ++$rm_counter == 1 ) {
                if ( has_post_thumbnail() ) {
                    ?><li class="rm_first_post">
                        <?php the_post_thumbnail();?>
                        <h3><a href="<?php the_permalink();?>"><?php the_title();?></a></h3>
                        <?php the_excerpt();?>
                    </li><?php
                } else {
                    ?><li class="rm_first_post">
                        <img src="<?php echo get_template_directory_uri();?>/images/wene.png" alt=""/>
                        <h3><a href="<?php the_permalink();?>"><?php the_title();?></a></h3>
                        <?php the_excerpt();?>
                    </li><?php
                }
            } else {
                ?><li class="rm_posts">
                    <a href="<?php the_permalink();?>"><?php the_title();?></a>
                </li><?php
            }
        endwhile;
        ?></ul><?php
        wp_reset_query();
    endif;
}

function rm_category_posts(){
    $rm_categories = get_categories();
    foreach( $rm_categories as $rm_category ) {
        $rm_args = array(
            'cat'                => $rm_category->term_id,
            'posts_per_page'    => 5
        );
        query_posts( $rm_args );
        if( have_posts() ): 
            $rm_counter = 0;
            ?><h2 class="rm_category_name"><?php echo $rm_category->name;?></h2>
            <ul id="rm_category_posts"><?php
            while ( have_posts() ) : the_post();
                if ( ++$rm_counter == 1 ) {
                    if ( has_post_thumbnail() ) {
                        ?><li class="rm_first_post">
                            <?php the_post_thumbnail();?>
                            <h3><a href="<?php the_permalink();?>"><?php the_title();?></a></h3>
                            <?php the_excerpt();?>
                        </li><?php
                    } else {
                        ?><li class="rm_first_post">
                            <img src="<?php echo get_template_directory_uri();?>/images/wene.png" alt=""/>
                            <h3><a href="<?php the_permalink();?>"><?php the_title();?></a></h3>
                            <?php the_excerpt();?>
                        </li><?php
                    }
                } else {
                    ?><li class="rm_posts">
                        <a href="<?php the_permalink();?>"><?php the_title();?></a>
                    </li><?php
                }
            endwhile;
            ?></ul><?php
            wp_reset_query();
        endif;
    }
}

<?php rm_category_posts(); ?>