When a person has heard the good news, is it by the work of God and the person’s
cooperation that they come to faith in Jesus?
Yes
If so, why are we involved in the process?
Because it is our will to be united with that of God’s. Our will to learn to love : this comes from believing that God is our hope and help.
Is it because the Holy Spirit works through us?
Yes.
Does God choose us to participate in His work in this way rather than inform people about the good news
God is in our dailiy life. He is truly present and active for those who believe. If you believe Him, you will see that all things are meant to be. But if not, all things are coincidences, then never God’s plan. To say it differently : those who believe God, they will follow God’s plan, the good things He has prepared for them since the beginning (their good works), the certainty of His Promise. Those who do not believe will not follow it (God’s plan, the good works), and they will fall into uncertainty of chances, just as they believe them.
Havin said this, yes God want us to participate in His works BY BELIEVING HIM, thus our works will become His works so that we become the salt of the world (to proclaim the good news by becoming salt), not mere talking of the good news.
by His power apart from us?
If you believe, you will be given power in you. God who is in you will work things out for you, in you, above you, and around you…
Regarding those (especially in centuries past) who have lived in areas where the good news had not yet been brought, is it our teaching that God does not (ordinarily) intervene?
God who created heaven and earth is omnipresence, He is everywhere and in all time. The heaven and earth is created by His Logos since the beginning . But when sin came to the world. we cannot differenciate His Logos from the other’s lies. Thus God came into our time and nature to verbally and naturally explicit about “The Logos” who is Jesus Himself, who is “The Good News”
Thus the answer to your question above is : God and the Logos has always been there since the beginning. Some wise people of the past may have caught some of His Logos as “wisdom”, but not explicitly face to face with God’s image (of which we were created with), thus their understanding was not complete, moreover due to their fallen nature, their beliefs were impure. Aside from that, God and His Logos always active, then now and forever.
His Logos is the Good News who is Jesus himself.
If so, do we have a teaching about this?
This is the teaching : God is faithful, He who created us has prepared for us all the good things from the beginning. His promise is His plan, and He who is faithful will surely fulfill His promise. We are given freewill to chose : to believe Him or not. Whoever believe shall be saved. Those who believe Him will do as He say, not by my own power but by The Spirit of God. The Spirit of God is the spirit of “believing and trusting God”.
Those who thinks that things are only of “chances” they will become unfaithful people and their lives are full of regrets. Regrets are of the spirit of unfaithfulness. But those who believe will know that God is almighty and trustworthy, thus He will keep them and surely will they see His good plan being fulfilled. And this is how to proclaim the good news : that we do as we say, not because we are strong holy people but because we belive in God’s love and forgiveness, so that we live as we believe, for all to see.