Hi,
This is easy to solve as it comes through your Joomla registration form. It is not in any way related to Acymailing.
What you need to do is to generate Google V2 invisible Captcha keys at Google, if you haven't done it already. Then add these keys to your Joomla plugin "CAPTCHA - invisible reCaptcha" and enable the plugin. Next go to Joomla->System->Global configuration, select "Users" in the left menu and make sure Captcha is set to "Invisible Captcha". Then you can enable your registrations again.
When you're in Global config, plase also take some time to check Contact persons in the left menu. Once there go to the tab Forms and check if the Captcha is enabled. If you don't use the forms feature disable it on this page.
As for not being able to update Acymailing, plese raise a support ticket via the contact form on Acyba's website.