Probably not. Very, very, very unlikely. An institution with 1000 years of reactionary history is not going to change. The teachings of Jesus are for the most part irrelevant in this discussion. As an institution the Catholic Church does not and has never taken them all that seriously. I would be less surprised to see Orthodox support for communism than Catholic, though both are unlikely, as i said, for historical reasons.

Yes communists should work with religious people and not be hostile to religion, but at the same time separating the religion from the inherently reactionary institutions like the Church. Religious people can be progressive and revolutionary, sometimes even individual clerics can be revolutionary, but institutions have their own agendas and raisons d'être, which are probably not compatible with our own.

I think official state secularism as in the Soviet Union and in the PRC is a good thing. It does not exclude people being able to privately practice their religion, so long as it does not interfere with the interests of the state.