ETA: Scratch that. Let's say we have two people, Bob and Alice. Alice is an atheist who lives a decent life and does no great harm to anyone, just minor sins here and there. Bob is a serial killer who's tortured and killed untold numbers of kids. On his deathbed, Bob accepts Jesus into his heart. Alice doesn't. What do Alice's and Bob's punishments look like?

This is an interesting, provoking, and common counter-example to the idea of mercy and acceptance of the Son—although it isn’t necessarily only facially applicable to Jesus’ forgiveness—and I understand where you are coming from here. I also used to think this way.

I would say, to be honest, that both would end up in heaven. Let me break down the general theory first and then address your questions directly.

1. I do not believe that one has to rigidly accept the Son of God (which may be Jesus if you would like) to be saved or that they have to participate in rituals (like baptism) to be accepted. As you alluded to with your example, someone can love God—love love itself: love goodness itself—without knowing the word “God”, having a concept of God that is robust, or having been exposed to some particular religion. God is judging us based off of our choices we make given the fact that we are not absolutely in control of ourselves (as natural organisms) and is evaluating how well we exhibited the virtues and, generally speaking, loved love (Himself).

2. For the vast majority of us, we have sinned before we die (although infants, e.g., haven’t if they are killed young); so for most of us we have offended God and, as I noted to @frank who ignored me, retribution is evaluated primarily based off of the dignity of the offended party (hence why shooting a rabbit illegitimately is lesser of an offense and deserving of less of a punishment than shooting a human the exact same way). With finite dignities, which are beings that are finitely good, there is a proportionate finite retribution (at least in principle) for every sin which one could, potentially, pay before they die (and thusly “serving their time” for the sin as it relates to the immanent victim—e.g., the human who was murdered). However, a sin is always also an offense against God and God is infinite goodness which is infinite dignity; so no proportionate retribution to something finite whatsoever can repay what is owed. This is why any sin, insofar as we are talking about the aspect of it that is an offense against God, damns us in a way where we ourselves cannot get out.

3. Loving love—being the a truly exceptional human being—will not repay the debt owed to an offended party with infinite dignity: Alice, or anyone of a high-caliber of virtue, is facially damned if they have sinned at least once.

4. God is all-just and all-merciful. He is all-just because He is purely actual and a creator, and so He cannot lack at anything in terms of creating; but to fail to order His creation properly is to lack at something as a creator. Therefore, God cannot fail to order His creation properly; and ordering His creation properly is none other than to arrange the dignity of things in a hierarchy that most reflects what is perfectly good—which is Himself. He is all-merciful because He is love and love is to will the good of something for-itself even when that something doesn’t deserve it. Mercy and justice, however, as described above, are prima facie opposed to each other: if, e.g., I have mercy on you then I am not being just and if I am just then I leave no room for mercy. To be brief, the perfect synthesis of the two is for a proper representative of the group of persons that has an appropriate dignity to pay the debt of their sins so that if they truly restore their will to what is right they can be shown mercy.

5. God must, then, synthesize justice and mercy by allowing a proper representative of humans to pay for our sins; but no human can repay it. It follows, then, that God must incarnate Himself as a human to be that representative. EDIT: I forgot to mention that God is the only one that can repay the debt because He is the only one with infinite dignity to offer as repayment.

6. The Son must be the one out of the Godhead that is incarnated because God creates by willing in accord with knowledge; His knowledge of Himself is what He uses to incarnate Himself; and the Son is His self-knowledge.

So, let me answer your questions with that in mind:

1. Alice and Bob have NOT committed equal sins: I don’t think that the fact that any given sin is unrepayable to God entails that all sins are equal. It just entails that all sins require something of infinite dignity to properly repay. Admittedly, it gets kind of weird fast working with retribution for infinite demerit. For example, in hell both of them will be punished for eternity but Alice’s punishment would be something far far less than Bob’s.

2. Since God saves us through His mercy (as described before), God does not have to punish us if we repent; and repentance is not some superficial utterance “I am sorry!” or, for your example, “Jesus I accept you!”. Repentance is normally through the sincerity of heart and through actions. A person who has never heard of God at all could be saved, under my theory, because they sincerely love love itself—God Himself—through action and this doesn’t need to be a perfect life that was lived (since God must sacrifice Himself to Himself to allow for mercy upon us). Alice, I would say, would be repentant in action and (most probably in spirit) for any minor sins she commits because she is such a good hearted person. If she were to do a lot of things that are virtuous but have the psychological disposition that doing good and loving her community, family, friends, etc. is horrible and something she despises; then she isn’t really acting virtuously. That’s like someone helping the poor as a practical joke or something instead of doing it out of love.

3. For Bob, it gets more interesting: your hypothetical eliminates the possibility of the good deeds part of what is normally a part of repentance since he is on his death bed when he has a change of heart. I would say that assuming he is not superficially saying “I am sorry (psst: hopefully I get into heaven this way!)”, then I would say that God’s mercy would allow him into heaven—at least eventually. Maybe there’s a purgatory faze where he is punished a bit for it first: I don’t know. However, what I do know is that Alice will be rewarded more than Bob; because reward is proportionate to the good deeds you have performed and goes beyond giving someone mercy from punishment. I do not believe that everyone in heaven is equal; or that God loves us all the same. That’s hippie bulls**t