Separation, church and state

To the extent that I understand this subject—which may not be much—is that some (folks) hold that church and state should be separate, each an institution with its own government so to speak.  

Is church and state separate; are these two institutions uniquely independent, each a self-governing institution with all its laws and bylaws, deeds and doctrines?  

I am sure that it is not, separate; but in fact, the state and church are inextricably intertwined whether fully intended, idealistic or not.   

Why am I certain, sure? 

Before giving cause or reason for my surety, certainty, some consideration of our history bears out the interrelationships of church and state.  Many churches display a national flag whether in public or within their sanctuary, yet also hold that brotherhood extends to all nations, the proclamation of their beliefs to other nations—even those not in good standing with the state.  

How can the church seemingly give allegiance to the state but then reverence to all of man, each and all sinners in God’s sight and by God’s word?  

What about us versus them; "us" the good and them the bad? 

I know that the subject is much more, complex and controversial, but what I am suggesting is that the church should render under to Caesar what is the state and unto God what is society, universal.  

If the church is a sacred house of God should it render allegiance to the state with a flag or by any other?  I cannot say, but only that the church and state are sometimes so close as to be indistinguishable let alone inseparable.