1) Please check and confirm the ‘Mailchimp action URL:’ in Newsletter content builder item.
2) Your test emails are going to your spam folder. If you use the same address of the recipient or use subject ‘Test’ sometimes these emails are sent to spam.
3) Your domain cannot send emails. Check with your hosting provider if you can send emails from your domain.
To check whether your domain can send emails set up an account at one of the three dedicated SMTP providers. All have free accounts options available. Each service has a plugin available on WordPress.org which you can install to setup even quicker or you can use the generic WP Mail SMTP plugin and enter your credentials.