And now we're finally at the meat and bones of the issue. I wasn't aware the "God that rewards" view was so ancient. I suppose if this belief was held by Tridentine fathers, I should be able to find mention of it in Trent? Where should I look?

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It never came up at Trent, since it wasn't an extant controversy.  As theologians, there were several Tridentine fathers who maintained this view, and according to Van Noort they did so precisely because they believe it better harmonized with God's mercy especially given the recent (for them) revelation that there was an entire half of the world that had been unexposed to the Gospel for a millennium and a half.  Not dissimilar (apparently) to the sort of thing Fellay and Lefebvre had in mind, as far as I can tell. 
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Although it is a different view, Cardinal Billot maintained that those who lived without the gospel were, without the gospel, such moral idiots that they were incapable of moral actions one way or the other and so while they did not merit heaven, they went to Limbo.  I bring this up as a way of showing that for hundreds of years, great theologians have experienced an impulse to see how God's mercy can be extended to those who truly did go without the Gospel.  It is not something that only crops up with modernism.
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Saint Augustine memorably held that those who do not hear the Gospel are damned to the pains of Hell, since faith comes by hearing.  He believed that those who truly never heard it did not hear it as a punishment for their sins.  Saint Thomas Aquinas also maintained the necessity of an explicit faith in the Holy Trinity and the Incarnation.  Given the weight of these two doctors, this seems to be the more credible view.  But the "Rewarder" view is neither condemned (yet), nor is it new, nor is it something that only modernists committed to.  

You think John 1:9 suggests that every single human being actually knows or is able to know about Christ and the Trinity?  How exactly do you reconcile that with sheer reason?  (BTW, I do not believe this verse says this.

It means that every person is born with a conscience.  It means all are born with the knowledge of God and the natural law.  Those that *try* to follow the natural law/10 commandments (i.e. have good will), will be given the graces to convert.  As Scripture says, God wills that all "come to the knowledge of the Truth." (i.e. knowledge of Christ).  
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To answer your question...is every human being "able" to know about Christ and the Trinity?  Yes, they are able, and God will provide, if they have good will, because God can read all hearts, which we cannot.  Those whom God does not give such graces to, He deemed that they would not accept, as only He can know.  To argue that a man lived his whole life desiring religious truth and God withheld it from him, thus damning him to hell forever, is a supreme blasphemy  (not saying you believe this, but many argue such, without thinking of the conclusions of their emotional complaints).  Let us not forget that salvation is a supreme mystery which no man can explain or understand because no man can read another's heart, nor know the inspirations/graces that God has given Him, nor the graces which than man rejected.

Well one, the text does say whoever believes and is baptized is saved, but it doesn't *necessarily* say that everyone who isn't baptized, will be damned.  

Yes it does.  The requirements are 1) belief and 2) baptism.  Christ said that those who do not believe are condemned, because obviously, if they do not believe, they will not be baptized.  And the reception of baptism without belief is null anyway.  

Secondly, this is why I included the 2nd quote, from John 3:  Jesus answered: Amen, amen, I say to thee, unless a man be born again of water and the Holy Ghost, he cannot enter into the kingdom of God.

This tells us what baptism is:  water and the Holy Ghost.  Nothing more, nothing less.  The Church has expounded on these doctrines and reiterated them many times in Her councils, which doctrines must be believed to gain heaven.

Two, the text has in view those creatures to whom the gospel was indeed preached. It seems, at the least, that *this* text does not address the fate of those who simply have not heard the gospel.  

Christ told the Apostles, "go and preach the gospel to the whole world."  Does God want people to save their souls or not?  Would God create requirements that were impossible to fulfill?  Is God a savage tyrant who creates souls and human beings, only to have pleasure at sending them to hell?
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Let us remember that salvation is a mystery.  Let us also remember that salvation is a gift from God; it is not a right.  Salvation is a choice to be made by all men.  ...a choice many times which is made in the interior soul, a choice made by the conscience.  A choice to believe or not to believe.  Can anyone measure this?  Can anyone have proof of the # of times that God beckoned a soul to Himself?  No.  How can we measure or know that individuals have not heard the gospel?  The gospel for the Israelites was simple - 1) believe in the coming redeemer and 2) follow the 10 commandments/natural law.  The pre-gospel for non-catholics then, is the same: 1) believe in God, 2) follow the 10 commandments/natural law.  All men, therefore, have the "gospel" on their hearts when they are born.  If they follow this gospel, God will give them greater graces and give them the True religion.  If they cannot follow the basic gospel, then for God to have the Catholic gospel preached to them is to "cast pearls before swine".

Three, it doesn't specify exactly what level of precision of "belief" Jesus is talking about.  Certainly to obstinately deny even one dogma leads to certain damnation, but also obviously knowledge of every dogma isn't necessary.  So what's the bare minimum threshold to count as "belief" in this passage?  The passage doesn't say.

Why would you expect it to?  You can't read 1 or 2 verses and expect it to explain everything.  That's the protestant mind coming out in you.  The basic doctrines required to receive baptism are the belief in the Incarnation (which is the fulfillment of God's promise to Adam and Eve for a Redeemer, and thus a completion of the Jєωιѕн religion) and a belief in the Trinity.

I suppose you couldn't repent of existing, but you could presumably have perfect contrition for any mortal sins you have, as well as the desire of baptism.

You're mixing and matching arguments.  We were talking about an "invincibly" ignorant person.  They could not desire baptism because they don't even know what it is.  They could repent of their sins, sure.  They would still be stained with Original Sin, so they could not gain heaven.  But they would go to Limbo and not hell.  
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It is said that though abortion is a great evil, since God allows sin because of free will, this means His permissive will allows abortion to occur.  However, for all those babies who are aborted, one saint said that this is an act of mercy from God because many of those babies might have grown up and be sent to hell.  Yet in dying innocently (except for Original Sin), they will go to Limbo, a place of natural happiness, and while this is not heaven, they would still avoid hell for all eternity.  What a just and loving God!

Because that's the way God set it up.  By analogy, a cancer patient *could* be healed directly by God, if God wanted to do that, but ordinarily medicines are used.  And God set up water baptism as the normative means to save souls from original sin and past mortal sins.  To spurn that just because God *could* do something else in an extraordinary case of inability is presumption, and is a good way to make sure you end up damned.

Your analogy doesn't work because the Church has never sanctioned the idea that people can get to heaven outside of baptism.  The Church is bound by Scripture and Tradition.  Nowhere in either is it taught, authoritatively, that one can gain heaven outside of water baptism and the Holy Ghost.
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Secondly, whether God can "bend the rules" is not up for debate.  Christ and the Church teach a,b,c are required and that's that.  Plenty of saints/theologians have speculated on the rule bending but they don't speak for the Church.  And I think such speculation is dangerous because we are trying to humanly understand the Divine salvific process.  And this is a recipe for questioning God, losing one's Faith and prideful anger at not being able to understand the mind of God.
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As God tells us in Scripture, which applies to all men:  "Before I formed you in the womb, I knew you."  What man is created that God does not know who is parents will be, or when in history he will live, or in what city or in what house he will take his first breath of life?  What man is created that God did not provide his intellect, his talents, his virtues and vices, his personality his hopes, dreams and fears?  What man lives on earth whom God does not sustain every second of his life, every breath he takes, every movement of every muscle and activity of the brain?  Does God create life and yet not know where this life is?  Does God create human beings out of love, yet not desire that these humans love Him in return?  Does God not create every man to return love for love, and to gain heaven?  
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Of course he does.  How does He do this?  We can never understand, and it is a dangerous job to try, as God told Job ...chap 38:
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You might argue that its only explicit, but baptism of desire *itself* is explicitly taught by the Council of Trent.  I'm aware of the ways people try to get around this, but they don't make sense.

No it isn't explicitly taught, it is mentioned in passing.  Desire for baptism in the context of justification/state of grace is mentioned.  It is not mentioned in the context of membership/salvation.

This isn't true.  While some hipsterdox (term for basically online Eastern Orthodox who exaggerate their differences with Rome) deny "original sin" a whole lot of Eastern Orthodox seem to agree with us on the substance of it.  Most Protestants, if anything, *exaggerate* original sin, in the sense that they start acting like we have personal culpability for eating the fruit in the garden imputed to us (which is the other side of their false imputation coin but I digress.)

The Eastern Orthodox have valid baptisms, so what they think about Original Sin is besides the point.  Protestants either follow Luther or Calvin on Original Sin.  Either it doesn't matter because we're all hopeless sinners or God will not punish us and reward us with heaven if we are good.
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We do have personal culpability for Original Sin, in the sense that it will send us to hell.  It matters not how you explain it, but the guilt associated with this sin to our souls is just as extreme as the guilt that Adam and Eve had.  We do share in their sin, as being their children, genetically and spiritually.

All that said I'm not completely convinced its that rare.  Mostly because a soul could have genuine repentance and sorrow for their sins, motivated by the love of God, even if they don't technically know what an "act of contrition" is.  The fact that they are repentant of their sins because they love the Lord would seem to be sufficient in that case.   And honestly, if they're really doing it motivated by *the love of God* the fact that they *don't* think its "necessary for salvation" would presumably not stop them anyway.

Ok, but sorrow for sins is possible for all men, because all men are born with a conscience and the natural law on their hearts.  But sorrow for sin does not equal baptism, nor does it equal salvation (unless you are a catholic).  Sorrow for sin means that God will forgive you, and that you continue on the road of life, on the road towards God.  How many non-catholics did God take their life after they repented, so that they would not go to hell and die in their sins?  So that they would attain Limbo but avoid hell?  I'm sure many, many people God had mercy towards in this way.
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If a non-catholic is sorry for their sins, does this necessarily mean they would accept the full Faith and become a Catholic?  Of course not.  How many protestants have I met who are wonderful people, who try to live a good life (as far as I can tell) but who hate the Faith?  A lot.  Does natural goodness necessarily lead to the acceptance of the True Religion?  No.  Therefore, the best chance these people have is Limbo.  They can never gain heaven because they would not accept the Church as a means of getting there.
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you said its heresy to say that a "non-visible member" could be inside the Church.  I don't see that.

It is not for me to prove this, as I have Christ's teachings in Scripture and Church Tradition and infallible doctrine that say otherwise.  It is for you to show me how one enters the Church silently, indirectly or accidentally.  Show me where any of the Church Fathers thought this was possible.  This false idea is born of Modernism and sentimentality.  It has gained steam since the 1800s and has corrupted minds on a grand scale ever since.  You are either in the Church or you are not.  You are either a member or you are not.  You are either baptized or you are not.  As Christ said, 37But let your speech be yea, yea: no, no: and that which is over and above these, is of evil (Matt 5).

So now you agree that there is no salvation outside the Church but you do not know what "outside" or "inside" the Church means.  You deny that to be a member of the Church requires belief in articles of divinely revealed truth?  You deny that to be a member of the Church requires the reception of the sacraments?  You deny that to be a member of the Church requires being a subject of the Roman Pontiff?  You deny that the Church is a visible society, or at least that the body and the soul of the Church are co-extensive?  These are not open questions as you claim for debate among good-willed Catholics.  They are dogmas, formal objects of divine and Catholic faith.  Heresy is by definition the rejection of Catholic dogma.  Do you need these dogmas cited for you?
 
If you cannot tell the difference between "outside" from "inside," then you most certainly do not know the difference between "up" and "down."  

Yes, I'd like a citation because all of the citations I'm aware of deal with being outside the Church.

And BTW, its not that I "now" agree that there is no salvation outside the Church.  I've always agreed with that.  That's never been the issue.  

“Ignorance,” says St. Thomas Aquinas, “is a punishment for sin.” (De Infid. q. x., art. 1.)
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Pius IX. said “that, were a man to be invincibly ignorant of the true religion, such invincible ignorance would not be sinful before God; that, if such a person should observe the precepts of the Natural Law and do the will of God to the best of his knowledge, God, in his infinite mercy, may enlighten him so as to obtain eternal life; for, the Lord, who knows the heart and thoughts of man will, in his infinite goodness, not suffer any one to be lost forever without his own fault.”
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To reconcile these 2 quotes with Scripture and doctrine, the only conclusion is that GOD WILL ALLOW NO MAN OF GOOD WILL TO DIE IN HIS INVINCIBLE IGNORANCE.  And before death, he will be baptized and join the Church. 
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This is how we must understand this doctrine.  This is why missionaries travelled 1,000s of miles on foot to convert the invincibly ignorant (though many cultures were also visited by saints who bilocated miraculously  and preached the Faith).  God has never condoned shutting down conversions or missionaries because invincible ignorance saves by itself. 

//God has never condoned shutting down missionaries or conversions//

And neither is anyone else here.  I could argue with the other stuff you’ve said, but the straw men are kinda annoying