Always On Windows Server Edition
Do you also need the Datacenter or Enterprise Edition of Windows
  Server 2012 or 2016 for Always on, or can you set this up properly on
  Windows Server Standard 2012 or 2016?


Windows Server Standard Edition is fine.

Starting with Windows Server 2012 every feature of Windows Server is available in Standard Edition.  In particular Standard Edition supports Windows Server Failover Clustering which is needed for some Always On capabilities.