If a criminal enters the church, he or she may be given a brief refuge, out of deference for the church.
Yes. It used to be that police did not enter the church out of deference and then later out of the concept of separation of the church and state. That is no longer the case. If a criminal enters the church, you can be sure, he or she may be given a brief refuge, again out of deference for the church. But ultimately the Police will enter, if they have to, to apprehend him or her. Of course, this largely depends on the type of crime committed. For example, a violent felon will be arrested almost immediately given the danger the person poses to the church and the surrounding community. The same is true for an undocumented alien who is smuggling drugs.
But if the alien is simply seeking refuge to avoid being arrested by the Immigration authorities, especially if they are women and children, some churches have given them sanctuary for a while. Under the current laws, to overtly offer sanctuary is in direct defiance of the law in the US.