You probably figured this out already, but can try adding content type to the class.
$controller = new MY_REST_Posts_Controller('post');
Where are the files located? In the theme folder or somewhere else?
If the your custom files are located in a css folder inside your theme you can try:
<link rel="stylesheet" href="<?php bloginfo('stylesheet_directory'); ?>/css/my_custom_style.css" type="text/css" media="screen" />
or you maybe you can add
@import url("css/my_custom_style.css");
in the main style.css of your theme.
You can try creating separate folders for every site in xampp htdocs and create a separate database for every site in phpMyAdmin. If you don’t want to create multiple databases you can define a different table prefix for every installation. The table prefix is defined with $table_prefix in wp-congif.php
