A few questions:
1) Since everyone is offered the choice, how many will of themselves choose God?
According to the Bible a "few". But according to the number John saw in Heaven it still is a number that no man can number. Scripture says there was a remnant under the Law and even so there will be a remnant today.
2) If none choose God of themselves, is not the Holy Spirit's work required?
Titus, chapter 2
"11": For the grace of God that bringeth salvation hath appeared to all men,
"12": Teaching us that, denying ungodliness and worldly lusts, we should live soberly, righteously, and godly, in this present world;
"13": Looking for that blessed hope, and the glorious appearing of the great God and our Saviour
The Spirit is always needed for man cannot save himself but must let the Spirit lead him. Jesus stands at the door of our heart and knocks and there are many factors, mostly fear of God, fear of dying and going to a devils hell, tired of the world, belief there is a God and the choice is always there and if you choose God the Spirit will lead you to repentance and then are you "born again".
The woman who had the issue of blood? She said within herself "if I can only touch the hem of His garment I shall be made whole and push her way through the crowd to get to Jesus and then she received virtue. That kinda gives her something in it doesn’t it. Now when she got to Jesus she was not saved it still took Jesus Himself to save her but when He saw her faith, He did save her.
3) What is the difference between one sinful being choosing and one sinful being rejecting? What causes the one to believe?
and a rabbit to chase...
They love darkness rather than light because their deed were evil and neither do they come to the light lest their deeds be reproved.
This Scripture explains both those who chose God and those who chose evil.
The only way to get rid of the devil is to have the fear of God in you.
4) What about someone born in a country where God knows that the gospel will never reach him? Does he have a choice?
Revelation, chapter 14
"6": And I saw another angel fly in the midst of heaven, having the everlasting gospel to preach unto them that dwell on the earth, and to every nation, and kindred, and tongue, and people,
Hebrews, chapter 8
"11": And they shall not teach every man his neighbour, and every man his brother, saying, Know the Lord: for all shall know me, from the least to the greatest.
The Book of Job, chapter 32
"8": But there is a spirit in man: and the inspiration of the Almighty giveth them understanding.
Titus, chapter 2
"11": For the grace of God that bringeth salvation hath appeared to all men,
"12": Teaching us that, denying ungodliness and worldly lusts, we should live soberly, righteously, and godly, in this present world;
"13": Looking for that blessed hope, and the glorious appearing of the great God and our Saviour Jesus Christ;
"14": Who gave himself for us, that he might redeem us from all iniquity, and purify unto himself a peculiar people, zealous of good works.
I think I answered them all to the best of my ability.