Yes, all emails sent through SimplyCast are mailed through our own server infrastructure. You do not have to use your own mail server to send your email marketing campaign.
Internally, our compliance team double-checks emails going through our servers to prevent blacklisting and ensure that spam doesn't damage the reputations of our clients (sometimes people will sign up and try to send it, which affects all of our customers as well). If a problem comes up, our developers can access the emails to see if there is a specific bug or issue, or if the issue can be fixed or worked around. Outside of our employees, however, nobody has access to your material.
If your campaign isn't sending for some reason, your first course of action should be to double check the sender address, as well as the activate screen in SimplyCast 360. If you still run into sending problems, contact our support team as soon as possible, using the contact page on our site, where you will find our phone number and a contact form.