Is it possible to install Developer Edition of SQL Server as the primary development server? Is it possible to install Developer edition of SQL Server on a Windows server environment as development server? Each developer will have a local copies of same SQL Server edition and will connect to this development server for data/object synchronization.

Also, if you can refer a link for a full description of this product.
Indeed this setup is possible. When the developers open Sql Server Management Studio, instead of connecting to a locally installed instance (not necessary given your requested setup, but I would still install it) then they would connect to the shared instance on the development server.

That is how our shop is setup.