Well, for the connection, I’m seeing an interesting error on our end now:
32700 - parse error. not well formed
This is usually caused by something misconfigured on the server, but it’s difficult to pin down exactly what. Can you ask your host if they can see anything in the error logs that might help? Also, I noticed that the site is running PHP: 5.5.38 ; although that version is still supported, WordPress now recommends PHP 7 or greater:
https://www.ads-software.com/about/requirements/
Can you ask your host if it’s possible to update that? Maybe it’ll solve the communication problem.
As for the email issue, Jetpack uses the server’s OWN SMTP service to send the share-by-emails. It sounds like your server might not be configured properly to send emails through the “wp_mail” function, which Jetpack uses.
Can you try installing the Check Email plugin and sending yourself a test email to see if it works? You can download the plugin here:
https://www.ads-software.com/plugins/check-email/
If you didn’t receive a test email, you could try using a plugin like the ones found here:
https://www.ads-software.com/plugins/tags/smtp/
Those plugins will let you configure your own mail server information that WordPress will use when it sends out emails from your site.
If the function appears to be working properly, it means that Jetpack’s emails are filtered out before they can reach your inbox:
* They can be filtered out by your server before they even get sent out; your hosting provider should be able to check their server email logs to find out if the emails were blocked.