Welcome to the Microsoft Q&A (Preview) platform. Happy to answer your questions.
You are correct. Since the Azure domain is not something you can get an official cert for as it is owned by Microsoft you need to setup a custom domain name for your cluster and map it to the one provided to you when creating your cluster.
As per your second question, there is nothing wrong with using Self Signed certs however it is not recommended for production clusters. For dev clusters there is no reason to pay for a certified cert.