I'm afraid your questions are not entirely clear to me.

#1: If you're asking whether Micro Focus Enterprise Server supports the IBM OSVS dialect of COBOL, then the answer is yes - there is an OSVS dialect setting for Micro Focus COBOL. For more information see the Micro Focus documentation here.

#2: I suspect we'll need a more specific question here, but you may find some answers in the IMS Support section of the product documentation.

I do believe that IMS DB Batch running in BMP mode is indeed supported, despite not being specifically mentioned in the link Michael gave.

But, "for what level of support", it's not clear exactly what you mean by that.  As Michael suggested, a more specific question will be needed to comment further.

The application is code in the OS Cobol and currently being executed on IBM Z/OS with IMS DB as the database and IMS/DC for the transaction processing. The batch jobs are executed in the BMP mode.

I am looking for migration of these applications as-is onto Enterprise Server and wanted to understand the migration process, complexity involved and support for these batch and online programs to be run as-is on the enterprise server.

This is potentially a very broad and complex question, so I recommend you contact Micro Focus Sales and ask to be put in touch with a migration specialist who will be able to help you assess your existing application and determine whether it can be migrated as-is, or if not what changes would be required.

They can also help you with case studies and questions about compatibility with older versions of zOS and IMS, which I see you also posted queries about.

You may also want to look at the press releases and resource center areas of the Micro Focus web site, which contain case studies and similar information. I believe the Intersistemi migration involved IMS; you might want to look at that one.