Nginx is a http proxy (that can additionally serve http requests).
Actually Nginx is a HTTP server that supports reverse proxy, it was written to address the 10K connection problem.
wordpress runs on php and apache. So you'd really only have to deal with installing the db and the weird things wordpress might need like mods for apache
Incorrect, I have a large client base using Nginx for Wordpress hosting without any issue. It simply outstrips Apache with resource utilization and performance.
502 can be caused by two things
1) If Nginx is configured as a reverse proxy, the 502 will be due to failure to communicate with the HTTP server behind it.
2) If Nginx is the HTTP server with a dynamic handler like php-fpm, it will be due to a failure to communicate with php-fpm.
You can determine which simply by checking if '.htaccess' is being used. If it works then you know you are being reversed proxied to another server, likely apache. If you are being reverse proxied ensure they have configured the proxy and HTTP server properly by forwarding the client's IP and using a module such as mod_rpaf or mod_remoteip to ensure it is logged and PHP uses it instead of the proxy server's IP. The easiest way to check is to run the following script:
<?PHP
echo $_SERVER['REMOTE_ADDR'];
?>
Be sure to test this under HTTPS also, checking that $_SERVER['HTTPS'] is also set as if they are terminating SSL on the reverse proxy PHP will assume no SSL unless they have catered for this.
Serving out the index.php file directly is a pretty big error, they practically just gave away the source code to your website. Since your website looks fairly simple, if you have any hard coded passwords in that file I suggest you change them ASAP.
If you are interested PM me, I am able to provide custom hosting of whatever you require, I own and operate servers in AU, EU and US that I can deploy to including cheap high performance VPS solutions with full root access.
Edit: Actually looking at your content if you can provide more information I would be interested in sponsoring your hosting.