I would say that a Christian is no longer a Christian when he ceases to profess the true religion.
Likewise, a church becomes a non-church when it ceases to profess the true religion. In the Bible, we see people who call on the name of the Lord being considered God's people, even in times of great apostasy and merely formal profession and hypocrisy. (I think this insight can be found in Bannerman's
Church of Christ.)
The profession of the true religion is the fundamdentals of Christianity: that which is essential to Christianity that makes it Christian and which lacking, becomes non-Christian. As is known, the exact limits of the fundamentals is difficult to determine. Certainly, the Trinity, deity of Christ, and the Incarnation are among them.
Rutherford (who held to the validity of the Papist baptism) viewed the church of Rome as a false church (he discusses this in
Due Right of Presbyteries). Rutherford said that Rome professes the true religion, but it teaches and practices so many contrary things so as to undermine and strike against the fundamentals, i.e., strikes against and denies the truth that it professes with its mouth. Or in other words, it professes the true religion falsely. Hence, it is a false church, not a non-church. Worse, it was once a true church, so it is also an apostate church. Worse still, it is headed by Antichrist. Rutherford said the next ecumenical council should excommunicate the church of Rome.
Nevertheless, Rome is not a non-church, and enough of the truth is professed so as to bring salvation to individuals within the Papist communion: as there is no ordinary possibility of salvation outside the visible church, so there is an ordinary possibility of salvation within the Papal church. Having saved them, the Lord will lead them out into a true church.
I would say that the fundamentals are those things contained in the catholic creeds; I don't recall if people like Rutherford said that though. These catholic creeds delimit the fundamentals of Christianity according to and with the authority of ecumenical church councils. Those who do not confess them are not within Christian bounds, even as those who do not confess a Reformed confession are not within Reformed bounds.
The same reasoning would apply to the PCUSA. So far as I'm aware, they still profess the true religion (holding to the catholic faith), but like apostate Israel of old they are worshipping false gods also. They are therefore a grossly corrupt and apostate false church, but not a non-church yet.
As for when some former church becomes a synagogue of Satan, it depends on whether we mean "non-church" or "false church." A church becomes a false church by a false profession of the true religion (professing fundamentals but so undermining and corrupting them to regard the profession as false). A church becomes a non-church by not professing the true religion at all. Thinking in terms of this language, with regards to its false profession, the church of Rome can be considered falsely constituted and therefore no church at all. However, the catholic visible church (to which the ordinances and ministry are given) extends beyond the bounds of particular churches, and the catholic visible church still exists within the church of Rome. So, due to this and due to Rome's profession of the fundamentals (and also its ministry and baptism, which were considered to have their essential form and so valid, since they are relative to the catholic visible church, not particular churches), it still retains the form of a church and is regarded as a church in that sense. Hence, you find people like Calvin and Turretin using this sort of language with regards to Rome (and very likely, WCF 25.5 too, since Antichrist is said to exalt himself in the church, and Rev 18:4 shows God's people were in it).
Note: In using the terminology, "true" and "false" church are judgments that can only be pronounced on particular churches. It is not a judgment that identifies whether a church or individual is a member of the visible catholic church or whether the visible catholic church exists within an entity. So one could say that Rome (a particular church) is a member of the visible catholic church (since it professes the true religion), but can say without hesitation or qualification that it is a false church. Because the ministry and ordinances are relative to the catholic visible church, it is possible for a false particular church to have a valid ministry and administration.
Quotation from Puritan Sermons or Morning Exercises (vol. 6, pp. 47, 48), by Peter Vinke. From an earlier
thread.
"WHAT WE THINK CONCERNING THE CHURCH OF ROME.
But more particularly as to "the church of Rome" (for so we call them that, professing to hold the Christian faith, are united in subjection and obedience unto that see, and do acknowledge the pope their universal pastor): when we call them "a church," we mean no more than that they are a society or company of men who make profession that they are Christians. Thus the Laodiceans are called "a church" (Rev. 3:14), though they were "wretched, and miserable, and poor, and blind, and naked," and we do not read of a sound part amongst them (verse 17). Thus God himself calls the ten tribes his people, after their defection, by reason of circumcision, which they yet retained, and their being the offspring of Jacob (Hosea 4:6). In this sense, soundness of faith is no more essential to a church, than health is to a man. And as a man that hath the plague or leprosy is still a man, though to be shunned; so they may be thus a church, though by all means to be forsaken. But as they themselves take a church for "a company of true believers joined together in communion," so they are no church, their faith being far from the faith of our Lord Jesus Christ."