What level of control do I have over my custom domain?
You own your domain and your original content.
You register and are responsible for paying and maintaining your custom domain. According to the Medium TOS, you also own the rights to whatever original material you publish on Medium.com.
You must have ownership or copyright privileges of content you publish on your custom domain. You cannot publish someone else's intellectual property or material, outside of Fair Use.
Our Copyright & DMCA Policy are still in effect, even though it's your custom domain.
Full Access To Medium Features
By using Medium to publish on your custom domain, you are able to access the full features of the Medium platform.
At this time, Medium only offers custom domains being mapped to Medium publications. You cannot map an @profile name to a custom domain. So if you have a publication you'd like to have linked to a custom domain, all you have to do is email us a request and we will help you iron out the technical details via email.
You can start by setting up your own publication on Medium and buying a domain name from a registrar of your choice.
You may want to read through our publications section to get to know how publications work and what features and functionality are already built in and available to you.
Though we are currently working on improvements to customization options for custom domains, for right now you will not be able to add any custom code blocks, widgets, HTML or any other code outside of what is available through Medium Publications functionality.