Why is faith the quality that God chooses to reward?

I think one reason that's supported by Ephesians 2 and 1 Corinthians 1 is that God made salvation by faith so that no one could boast. If it were by works or by human understanding, people could boast--or even if the faith were by ourselves. But our saving faith is a gift from God. So the way God worked salvation for those who believe in Jesus glorifies Him.

Also faith is a key part of love. It's true of human relationships and it's also true of our relationship with God. See e.g. Proverbs 3:5-6 and Hebrews 11:6 (note that to please God we must believe in his Son as other Scriptures say--that's not all spelled out in this one verse, but is elsewhere). And as others have already said, we aren't and can't be righteous, in addition to the fact that salvation by works would be grounds for boasting as Romans points out.

I believe the premise of this question is a bit off. I think I know where you're going with the idea that all Christians believe that "God rewards faith" but a good chunk of the Protestant/Catholic divide was fought over this and it's not because we believe the same things but we just call them different things.

To Catholics, there are two important scriptural points that make it clear to us that God values "Faith and Works". One, is from the book of James, where he says that faith without works is dead and the other is the stickier point in Romans where St. Paul is more or less trashing good works, but Catholics interpret that as "works of the law" and Protestants as "works in general".

Now, there are a plethora of examples throughout scripture where God +1's a faithful act and -1's an unfaithful act. Starting with Abraham all ready to sacrifice Isaac. That's a +1 (and he got to keep his kid); then Moses tapping a rock twice -1; Everybody who was frightened of what the spies reported in canaan -1 (back to the desert with you). Israelites grumbling and worshipping idols -2, Israelites looking on a serpent on a stick and being healed +1, David believing he can defeat goliath with stones and faith +1, David thinking he can take another man's wife -1.

In life, God rewards faithful acts and curses people (Moses and nearly all the Israelites who left Egypt never got to see the promised land, David's sons were losers, Aarons son's were killed, Aaron and Miriam became lepers, then they didn't)

But, then there are some things that aren't as cut and dry, Job, was punished for no good reason, his kids were losers, but them getting killed wasn't his only punishment. His wife was punishment enough, but even his friends were jerks. All the boys in Maccabees were killed and they kept the faith. Daniel and his buds were not killed and they kept the faith. Esther had everybody pray and asked very sweetly and she was granted deliverance, on the other hand Deborah snuck around and lopped off the head of her oppressor. One thing is for sure, that all the pagan priests who tried to screw around with righteous Israelites got their comeuppance.

Peter was literally called "Satan" and "ye of little faith" by Jesus and he's esteemed among the greatest of saints and his office passed down to his successors is vicar of Christ on Earth.

But you're not asking about theodicy or a synopsis of the Bible. You're asking why does God do something and all I can answer is that God does whatever He wants. The saints will tell you this (especially St. Alphonsus), that God wants us to seek His will. Jesus says that too "seek ye first the Kingdom and the Will of God". Faith, being "the substance of things hoped for, the evidence of things unseen" is wrapped up in obedience to a God you cannot see "Blessed are those who believe and do not see". That has to be rewarded because it doesn't exist any more in heaven, it's entirely cashed in for the beatific vision where Faith and Hope aren't needed any more; but love is.

As to reward, perhaps you're referring to ones "place in heaven", the idea might be original to Catholicism, or other denominations might not have a doctrine around it. But there's an idea that everyone will be perfectly happy in Heaven, but individuals will have a greater or lesser degree of glory. I can't tell you what that means, but I'd be happy to read another question on this site asking about it.

On the other hand, Jesus says that not everyone who says "Lord Lord" will get in, and if you do get in you've gotta wear a wedding garment to the banquet or else you'll get thrown out where there's wailing and gnashing of the teeth. So, sort of implies doing something rather than just believing something, but we believe that God knows the rationale for the doing is the believing so it all comes down to faith (and works).

God is righteous and none else is.

And this is the answer to your question.

God demonstrated his righteousness upon his own Son, righteously laying sins upon him (which were not his own) and righteously imputing sin itself to him (again, which was not his own) that by suffering 'in the body, on the tree' he should bear away sins and through death he should eradicate sin.

Those who see and believe this righteous demonstration are seeing and believing the righteousness of God in demonstration.

God therefore sees - in their faith - his own righteousness.

'Abraham believed God and there was evaluated to him unto righteousness' is the literal and accurate translation of the Greek in the Septuagint (Genesis) and of the Greek in Romans, Galatians and James.

God sees and values that within the faith of the believer - which is his own righteousness.

Thus the reason God evaluates faith (that particular kind of faith which appreciates his own righteousness) is because he is righteous.

No other response would be right.

The Bible states that, "Without faith, it is impossible to please [God]; for he that cometh to God must believe that he is, and that he is a rewarder of them that diligently seek him" (Hebrews 11:6 KJV). That whole chapter lists many people of faith, right from Abel onward. The question you ask is, 'Why does God choose to reward [such] faith?'

But your second and third assumptions are strange, and not necessary to answer the question. Faith, you see, is not about measuring our character. Faith is not about us. It's all about God. It is WHO we have faith in that is the crunch point - not that we have faith!

Your third assumption is odd, because even if I did not think 'things are going more or less to plan' I could still have faith in God, that he would achieve his plan despite everything appearing to be going haywire, due to faith in his power and timing. Also, with this assumption, 'free will' has got nothing to do with it. Just asking your question with assumption 1 is fine; the other two only confuse matters.

You asked, "Why is God selecting the faithful instead of the righteous for salvation?" then added, "why is faith something that God ordained to be righteous?" I answer from a Reformed Presbyterian position. The Bible says "The righteous will live by faith" (Galatians 3:11). Also, Romans chapters 3 to 6 say a great deal about faith and righteousness. I cannot go through all of that, but can only give a brief summary.

Chapter 3 starts by quoting the Old Testament where it states that there is no-one righteous, no, not even one. No-one does good, either. Yet, despite that, God's righteousness is by faith of Jesus Christ to all who believe (3:20-23). The point is made that all have sinned and fall short of the glory of God, but God will justify freely by his grace through the redemption that is in Christ Jesus (vs. 24), through faith in his blood "to declare his righteousness for the remission of sins that are past, through the forbearance of God, to declare his righteousness, that he might be just and the justifier of him which believeth in Jesus" (vss. 25-26).

Now comes the reason for faith in Jesus being essential, in verse 27 - "Where is boasting then? It is excluded" by the law of faith, not the law of works. Abraham is then given as an example of how to please God, by faith. He showed faith (trust) in God before doing anything. He left Ur, trusting God's command to go to a place (he knew not where). He trusted God's promise of inheriting a land, and having a child in his old age that would require a miracle. From that child of promise would come the nation to inherit that promised land. Abraham had hope in God, against all hope, for all of that. That's all in chapter 4, a key verse being that the promise was not through the law, "but through the righteousness of faith" (vs. 13).

Chapter 5 verse 1 is the basis for your 1st assumption. Verses 17 to 18 sums up the reason for God counting faith as righteousness. By one offence death reigned by one, but those who receive an abundance of grace and the gift of righteousness shall reign in life by one, Jesus Christ. The offence of sin brings judgment of condemnation upon all men, yet God chose to show grace to undeserving sinners by judging Christ in our place, punishing him instead of those who put faith in Christ. God justly dealt with sin at the cross, which is why faith in Christ's shed blood is counted to us as righteousness as that is our way of declaring the righteousness of God. The last verse in chapter 6 concludes:

"For the wages of sin is death; but the gift of God is eternal life through Jesus Christ our Lord."

Nobody needs faith to believe that sin leads to death! But only faith in Christ's shed blood speaks to a faithless world of the gift of eternal life, which is unmerited. God is pleased when we trust in what he did, in Christ, to undeservedly do away with our sin. Such faith is all about God and Christ, you see. It's not about us, or about free will, or works. There can be no boasting, or pointing to ourselves as having faith, for it is all of God and must bring glory to God.