Moral implications of heaven and hell doctrine

The answer to your question "Isn't this a reductio ad absurdum, showing that the heaven/hell doctrine provides for no moral incentives for unbelievers to act good without Christian belief?" is (essentially) yes.

Christianity is more than doing good to others. That is the second commandment of the law. The first and greatest is to love God. (Matthew 22:36-40).

Hosea 6:6

For I desired mercy, and not sacrifice; and the knowledge of God more than burnt offerings.

This may run contrary to your notion of "good", but be that as it may, Christianity holds faith in God as necessary to true goodness, or righteousness.

Hebrews 11:4

By faith Abel offered unto God a more excellent sacrifice than Cain...

It wasn't because of his wisdom, charity, wit, etc. that Abel's offering was more excellent; it was because of his faith (in God).

That said, this statement is a bit leading to a Christian: "No matter how good an unbeliever behaves [he gets] the same exact fate."

The fallacy here, is the possibility people can be very "good" but choose to reject Jesus (and by extension, God himself).

All good comes from God. And therefore, good things acknowledge and worship God.

James 1:17

Every good gift and every perfect gift is from above, and cometh down from the Father of lights, with whom is no variableness, neither shadow of turning.

Likewise, those who truly love and worship God are good.

John 4:19

If someone says, "I love God," and hates his brother, he is a liar; for the one who does not love his brother whom he has seen, cannot love God whom he has not seen.

Now, it is important to remember than spoken belief (e.g. "I murder and have no remose, yet I believe in God") is not equivalent to true faith.

Matthew 7:22-23

Many will say to me in that day, Lord, Lord, have we not prophesied in thy name? and in thy name have cast out devils? and in thy name done many wonderful works? And then will I profess unto them, I never knew you: depart from me, ye that work iniquity.

Thus, true faith in God and true goodness, according to Christian belief are not separable. Of course, not all are taught Jesus, or at least not fully. There are several questions about the fate of the ignorant. (1) (2)

In some ways you properly describe the despair associated with hearing the gospel but not believing it. The problem encountered is that unbelief makes the gospel seem foolish.

For the message of the cross is foolishness to those who are perishing, but to us who are being saved it is the power of God. (NIV, 1 Corinthians 1:18)

One of the main reasons why unbelief finds the gospel intolerable is that it seems unfair to judge humanity based upon the works of one man. Why should little old ladies die because Adam sinned? In the same token it seems unjust that immoral killers, rapists, thieving drug addicts, etc. should inherent eternal life based on the work of one man, Christ. However this is the gospel.

The primary objection that unbelief poses is self-righteousness. Unbelief says 'Eternal judgment for all humanity for one sin, is too much!' Not knowing how evil evil is. It also barks, 'if Adolf hitler cried out to Christ in faith on his death bed, he would not deserve to live in blessed glory! That's too much grace!'.

The only real answer is not a philosophical one, but a theological one. God is wise and strong, we are dumb and weak. The apparent foolishness of the gospel is stronger then all our wisdom. Our conscience knows this, that is why we are bothered, so it is best in the interest of ourselves to cling to Christ and deny human goodness as a means of eternal well being.

There is one element though that does not seem to be understand in the despairing sentiments posted. Why 'faith' as the only way? Upon inspection, we can see God's wisdom in this means because only by faith can a person receive something without doing anything morally right. It is a method that a man without any good morals, can receive something good, while being only a sinner. Any other requirement might include some kind of moral effort, but faith in Christ does not require any moral effort. This is why Christ is often called a gift. This concept might at least begin to brighten up the dismal sentiment described. Especially when one considers this free, eternal unfathomable blessing to be absolutely a simple gift from a loving, omnipotent God. Then the extremity of punishment for sin is brought into harmony with the extremity of his glorious willingness to declare sinners holy and free.

These are eternal matters that can't be comprehended with human reasoning but are revealed to us through the scripture alone.