Q:

Does IBM bluemix have some sort of incoming or inbound email service or APIs?

Does bluemix (https://console.ng.bluemix.net/) have any incoming mail system for their apps which is similar to Google App Engine's Receiving Mail system (https://cloud.google.com/appengine/docs/python/mail/receivingmail). 
Thanks.
PS: If possible, please share some links so I can begin exploring!!

A:

as already suggested on the previous response Bluemix provide a outgoing email service using SendGrid
https://console.ng.bluemix.net/catalog/sendgrid/
For incoming email service, there is no service provided on Bluemix, but there isn't any limitation in integrating your application on Bluemix with an external email service, using its own POP/IMAP interface
How to integrate your app with an external email service is depending from the language/technology you wish to use, for example with PHP you could use its IMAP extension
http://php.net/manual/en/intro.imap.php
to read from an external service using IMAP or POP3
Moreover using an external email service may allow you to use that service for sending email through SMTP, instead of using SendGrid: for example using PHP runtime you could use 
https://github.com/PHPMailer/PHPMailer
to integrate with an external SMTP service (usually available with an email account as well as the POP3/IMAP one)

