If what we are doing is just the fruit of having first been saved, then it is something different than our salvation, but it says to work out our salvation. In Titus 2:14, Jesus gave himself both to redeem us from all lawlessness and to purify for himself a people of His own possession who are zealous for doing good works, so that is also the ongoing aspect of our salvation that is in accordance with what he completed through the cross.
You missed the next verse. God moves within us so that our working will lead to, not to becoming saved, but to a growing in belief, and understanding of salvation. IOW, it doesn't matter where one's understanding or faith begins from, if one of the elect, God will move them to a correct understanding of Christ - God is only within those who already have already become saved.
[Phl 2:12-13 KJV]
12 Wherefore, my beloved, as ye have always obeyed, not as in my presence only, but now much more in my absence, work out your own salvation with fear and trembling.
13
For it is God which worketh in you both to will and to do of [his] good pleasure.
We have already explained that good works follow, and is a result of, salvation, not the reverse.
It doesn't say that Abraham was counted as righteous by Christ's faith, but that Abraham believed God and that was counted to him as righteousness. The way to have faith in Christ is by obeying God's instructions and Christ expresses his faith by obeying God's instructions, so faith in Christ has the same content as the faith of Christ, which means that you are making a moot point. The one and only way that to become righteous is by faith and what we are becoming when we become righteous is someone who practices righteousness in obedience to God's instructions through faith. In Habakkuk 2:4, the righteous shall live by faith and in Isiah 51:7, the righteous are those on whose heart is God's law, so the righteous living by faith does not refer to a manner of living that is not in obedience to God's instructions.
No, it doesn't say that. It says Abraham was reckoned faith (belief) FOR righteousness: righteousness through Christ's faith. Christ's righteousness and his faith are one. When His faith is reckoned, His righteousness is too. Righteousness only comes with Christ's faith. That is what "from faith to faith" means: from Christ's faith unto faith within us. Man's faith has no righteousness in it, only Christ's does. Did you see/understand the verses I posted regarding righteousness? "the righteous" of Habakkuk 2:4, are those saved who were made righteous through the faith (and righteousness), of Christ reckoned to them - Christ alone is the Saviour, man is not, as such, if we then live, it can only be though/because of Christ and not because of ourselves.
Regarding Isiah 51:7, those who hearken to God are those who already know righteousness. To know righteousness comes by, and through, having been saved and having received the righteousness of Christ. The law of God is Christ Himself, which is the law of the Spirit of life in Christ Jesus -it is not the law of works, which is the law of sin and death.
Why would someone choose to obey "God's instructions" as you say, if they didn't already have faith to make them want to do so?
They would have no desire to, so would already have faith, in order to desire to acquire faith- a logical impossibility/ absurdity.