When two-factor authentication is first setup, you're prompted to setup backup codes which are to be used in the event you're unable to authenticate with your application.
Many provides, including Amazon, use a similar form as well as similar backup methods that allow you to get back in. It is end-user responsibility to maintain those codes and keep them safe for if or when you're locked out due to application error, wipe, or otherwise.
In regards to support itself, DigitalOcean is completely un-managed unless you're a business client, which means ticket-based support is the only option available. The exception to this is if you are a business client (spending more than $500/mo) -- in such a case, you'd need to have your account status changed -- which would open up additional means of support.
When it comes to ticket-based support, it's provided best effort from what I recall. They will help, but when it's user-error or not related to network, data center, hardware, or platform, support is limited.
This isn't just specific to DigitalOcean, any provider that provides un-managed service works in a very similar way. This includes Amazon, Google Cloud (GCP), Linode, Vultr, and various others.