In Islamic countries, what is the relation between the religion and state of affairs?

Yes and no. This isn't an easy one to answer. Islam means "submission" (to Allah) so there certainly is an authoritarian side to the religion, but that is no different than the Catholic Church in Europe. In addition, some of the finest scholars and technological developments were borne in Islamic Countries during the "Middle Ages," so that belies their present state of affairs. Islam used to be very accepting and progressive in the sense Westerners conceive it today.