OK perfect.
Azure Active Directory Domain Services is just a standard Active Directory forest where you don't need to manage the servers. But you will have nearly the same possibilities as a standard Active Directory domain that you would have deployed locally inside your datacenter.
However, Azure Active Directory and On-Prem ADDS are really different components. You will not for example be able to setup a trust between your Azure Active Directory and your AADDS. So, there is nothing logical to name your AADDS with a specific domain name that would be a sub domain. However, you should ask yourself the question about using AADDS OR deploying IAAS VM and deploy manually your Active Directory domain. Why you're needed an on-prem domain ? Maybe for some application that cannot be integrated directly through Azure Active Directory ? Is there any specific reason (just to be sure to understand your constraints).
Anyway in both scenarios AADDS or manually deployd ADDS, you can chose a complet different domain. But if you need after to get your accounts populated in your Azure Active Directory - you will require standard Hybridation process with AAD Connect. So if the UPN extension does not correspond to the name you've choosen for Azure Active Directory and O365, you will need to modify the UPN Suffix. Something that a lof of companies are doing when going to O365.