One can create, enable and disable a mailbox using Powershell commands. Is it possible to do the same using the EWS Managed API?
No, EWS is just a web service it has nothing to do with mailbox creation.
Here you can find all operations supported by EWS. [Exchange server 2007] http://msdn.microsoft.com/en-us/library/exchange/bb409286%28v=exchg.80%29.aspx
[Exchange server 2010] http://msdn.microsoft.com/en-us/library/exchange/bb409286%28v=exchg.140%29.aspx
[Exchange server 2013] http://msdn.microsoft.com/en-us/library/exchange/bb409286%28v=exchg.150%29.aspx
You can write c# application for Create,enable,disable mailbox using powershell