>I have switched the SQL Server 2012 to run in SQL Server 2012 instead of the emulation to 2008. So far, having removed the emulation, is being reported by two users that the impression is that this is very fast. Is it the case that it should be faster?

That's unusual result. Do you have some metrics that confirm the observation?

I haven't heard about speed improvement by switching the compatibility mode, but I guess it's possible.