One of the questions involved in God choosing who will be saved is who Christ died for. Now it should be noted from the outset that the value of Christ's death is infinite. But we do find Scriptures that tell of Christ dying for the world or all men (1 Timothy 4:10; 1 John 2:2; John 3:16-18; 6:33). Other verses, though, teach that Christ died for His people (John 10:11; 15:13; Acts 20:28; Ephesians 5:23-27). I would conclude from this that Christ's death is sufficient for and offered to all people, but it is effective for those who believe, who will ultimately be those God has chosen. I would see both types of Scriptural statement as true from their own perspective.
In this I would take the offer of salvation to all who believe as a legitimate offer (Acts 16:31; Romans 4:4,5; John 6:29). But we are sinners and cannot come to God (Romans 3:11, 8:8; 1 Corinthians 2:14) unless God does a work in us (John 6:44; 1:13; Ephesians 1:4-6). In the same way, the promise of salvation to those who keep all God's commandments is a legitimate offer, but none of us live up to it, and therefore we need Christ (Galatians 3:10; Romans 7:10-12; 3:19,20). Also, the atonement is not automatically applied to anyone; it is by faith, and until we are brought to faith we are still under sin and God's wrath (Ephesians 2:1-3; Colossians 2:13, 1 Corinthians 6:9-11). Salvation is unmerited, but it is not unconditional; it is on the condition of faith (Romans 3:28; Philippians 3:9), but God supplies the condition (Ephesians 2:8,9; Acts 13:48). Christ died for those believe in Him, but God chooses who that will be (Romans 9:16; 2 Timothy 1:9).
Does God then choose who will not be saved? That depends on what is meant. All things that happen (Ephesians 1:11), including the fate of the unbeliever, are under the control of God (Romans 9:22,23). But in God's choosing of the elect, there is a direct intervention of God, resulting in their doing what they would not naturally do (Romans 3:11; John 6:44: 1:13). This is similar to a miracle, rather than a natural event (though both are under God's control). Therefore, God is not responsible for the sinner's sin, but is responsible for the believers faith, as He creates it directly. Not that I claim to understand this philosophically, but it is what I conclude the Scripture teaches.
But we must use caution in trying to understand the hidden counsels of God (Romans 11:33-35; Isaiah 55:8,9). As John Calvin pointed out, those who try to pry into the secret thoughts of God will become trapped in an inextricable labyrinth. What God would have us know is written in His word, Calvin continues; with that let us be content. But I have a problem with any viewpoint which must take Scripture outside its natural meaning to support itself. The right viewpoint must take all of Scripture into account.