I don't think you can create an Essentials edition of Windows Server in Azure today. I checked and it's not available in the templates on Azure Subscribtion. However, it's not a big deal, if you agree with the pricing of a VM in Azure, the licences cost is integrated so no surprise on this. The template will be Standard or Datacenter.
Sure you can go to the cloud ! There is nothing that is preventing you from migrating to Azure. A lot of customers are doing so and want to keep their Active Directory Domain Services in this case they are "extending" their datacenter to Azure thanks to VPN S2S or Azure ExpressRoute. In this way, Azure will appear as an extension of your own personal datacenter. VM on-prem and VM in Azure will be able to speak together.
So in this case, you just need to create a new virtual machine IAAS in Azure and you will just manage it the same way as you do locally. You go in Server Manager and you install the ADDS Role. And it will replicate for an existing Domain Controller.