Wordpress - Changing Admin Menu Labels

Here's the process to change the labels (I changed posts to "contacts" in my example)

function change_post_menu_label() {
    global $menu;
    global $submenu;
    $menu[5][0] = 'Contacts';
    $submenu['edit.php'][5][0] = 'Contacts';
    $submenu['edit.php'][10][0] = 'Add Contacts';
    $submenu['edit.php'][15][0] = 'Status'; // Change name for categories
    $submenu['edit.php'][16][0] = 'Labels'; // Change name for tags
    echo '';
}

function change_post_object_label() {
        global $wp_post_types;
        $labels = &$wp_post_types['post']->labels;
        $labels->name = 'Contacts';
        $labels->singular_name = 'Contact';
        $labels->add_new = 'Add Contact';
        $labels->add_new_item = 'Add Contact';
        $labels->edit_item = 'Edit Contacts';
        $labels->new_item = 'Contact';
        $labels->view_item = 'View Contact';
        $labels->search_items = 'Search Contacts';
        $labels->not_found = 'No Contacts found';
        $labels->not_found_in_trash = 'No Contacts found in Trash';
    }
    add_action( 'init', 'change_post_object_label' );
    add_action( 'admin_menu', 'change_post_menu_label' );

To change the menu order, go with this:

// CUSTOMIZE ADMIN MENU ORDER
   function custom_menu_order($menu_ord) {
       if (!$menu_ord) return true;
       return array(
        'index.php', // this represents the dashboard link
        'edit.php', //the posts tab
        'upload.php', // the media manager
        'edit.php?post_type=page', //the posts tab
    );
   }
   add_filter('custom_menu_order', 'custom_menu_order');
   add_filter('menu_order', 'custom_menu_order');

I've got code to remove items, but it's globally and not based on user access level

To rename default post type (or any other for that matter) just use filter post_type_labels_{$post_type}. For default post it will be post_type_labels_post. In the code below is the full list of labels (WP 4.7.1). You do not have to change everything.

add_filter( 'post_type_labels_post', 'news_rename_labels' );

/**
* Rename default post type to news
*
* @param object $labels
* @hooked post_type_labels_post
* @return object $labels
*/
function news_rename_labels( $labels )
{
    # Labels
    $labels->name = 'News';
    $labels->singular_name = 'News';
    $labels->add_new = 'Add News';
    $labels->add_new_item = 'Add News';
    $labels->edit_item = 'Edit News';
    $labels->new_item = 'New News';
    $labels->view_item = 'View News';
    $labels->view_items = 'View News';
    $labels->search_items = 'Search News';
    $labels->not_found = 'No news found.';
    $labels->not_found_in_trash = 'No news found in Trash.';
    $labels->parent_item_colon = 'Parent news'; // Not for "post"
    $labels->archives = 'News Archives';
    $labels->attributes = 'News Attributes';
    $labels->insert_into_item = 'Insert into news';
    $labels->uploaded_to_this_item = 'Uploaded to this news';
    $labels->featured_image = 'Featured Image';
    $labels->set_featured_image = 'Set featured image';
    $labels->remove_featured_image = 'Remove featured image';
    $labels->use_featured_image = 'Use as featured image';
    $labels->filter_items_list = 'Filter news list';
    $labels->items_list_navigation = 'News list navigation';
    $labels->items_list = 'News list';

    # Menu
    $labels->menu_name = 'News';
    $labels->all_items = 'All News';
    $labels->name_admin_bar = 'News';

    return $labels;
}

If you want internationalization support, just use __( $text, $textdomain ) like:

$labels->name = __( 'News', 'textdomain' );

I found the filter in function: get_post_type_labels() from a file wp-includes/post.php:

/**
 * Filter the labels of a specific post type.
 *
 * The dynamic portion of the hook name, `$post_type`, refers to
 * the post type slug.
 *
 * @since 3.5.0
 *
 * @see get_post_type_labels() for the full list of labels.
 *
 * @param object $labels Object with labels for the post type as member variables.
 */
$labels = apply_filters( "post_type_labels_{$post_type}", $labels );

I agree.. The functions.php file gives lots of flexibility. I've needed some of the same functionality that you described with a combination of functions.php filters and this plugin.

From what I can tell..this plugin would accomplish both of your issues and it works well in Multi-Site install situations too. Hope that helps.