I’ve found a cache of your homepage from 10 Nov which shows you had 3.9.2 on that date. Can you be sure you had 3.9.3.
If you are unsure about an email you can check its headers. In Windows Live Mail, control-F3 brings up the headers and you can usually see who the email really came from. Other email clients may have different methods for seeing email headers.
Anyway, your site is giving a 500 server error which could be several things. I’ve checked 3 other sites on your server and they are working, so it looks like the problem is related to your site and not the webserver.
You need to get access to your site using FTP or your hosts file manager program. I would start be renaming the wp-content/plugins folder. This will put all plugins out of use.
If the site now works, reinstall the plugins one-by one to see which one is causing the error. If the site doesn’t work, try updating WordPress manually. Don’t lose wp-config.php. Also try uploading a fresh copy of your theme.
Also, check the main WordPress folders for files called error_log. This plain text file may contain clues. If you have cpanel, that also may have some log files. Consider asking your host if they find the error, though a lot of hosts don’t support user software.
You could also temporarily rename .htaccess to see if that gets you back up. WordPress will remake .htaccess if you reset permalinks.
I hope you have a recent backup, but if not you cmight be able to scrap your text and images from Google’s page cache in case it comes to the point where you need to remake your site.