It is indeed possible that Gmail filters all emails coming from your server because someone on your server was, or is still sending emails that Gmail consider as spam.
Can you try installing the Check Email plugin and sending yourself a few test emails, to different email addresses, to see if it works? You can download the plugin here:
http://wordpress.org/plugins/check-email/
If you didn't receive a test email or fail to receive emails when sending to Gmail addresses, you could try using a plugin like this one to configure your WordPress site to use a specific email service:
https://wordpress.org/plugins/postman-smtp/
Let me know if this helps!