Hi,
I was able to access your backend now. However, I'm still not able to test anything.
Note that the Captcha will only display if you're not logged in. If you're logged in, we suppose that you already registered and already went through the captcha check at that time. So the captcha won't display.
However, to access your frontend, a login is required as the website is in maintenance. And the backend access doesn't allow me to access your website configuration to deactivate the maintenance.
So in short, it's normal that I don't see the Captcha on the contact form on your test website, even with the plugin enabled and properly configured, and I can't access the registration page either since I'm already logged in.
Also, note that in order to check the different settings, I will have to access your Joomla configuration, the user manager settings, etc.
It would be great if you could make a super admin access and put a htaccess authentication on the whole backend AND frontend with strong passwords so that we can review the situation properly.