Fix Woocommerce and Email Errors

Woocommerce on my Wordpress website
Unable to send emails to customers
Unable to send emails to admins on customer sales
Payments being accepted when product stock not available.

Here is the error in regards to test emails via WP SMTP Plugin
There was a problem while sending the test email.

An issue was detected.
This means your test email was unable to be sent.

Typically this error is returned for one of the following reasons:

Plugin settings are incorrect (wrong SMTP settings, invalid Mailer configuration, etc).
Your web server is blocking the connection.
Your host is rejecting the connection.
Recommended next steps:
Triple-check the plugin settings and consider reconfiguring to make sure everything is correct. Maybe there was an issue with copy&pasting.
Contact your web hosting provider and ask them to verify your server can make outside connections. Additionally, ask them if a firewall or security policy may be preventing the connection - many shared hosts block certain ports.
Note: this is the most common cause of this issue.
Try using a different mailer.