Yes, salvation is totally dependent upon God (Rom 9:16). Yes, people cannot be regenerated by their own wills (Jer 13:23; John 1:12-13). Yes, people are regenerated by the will of God (Rom 8:29-30; 1 Pet 1:3). Yes, regeneration is an act of God upon the spiritually dead (Eph. 2:1,5). Yes, spiritually dead people can neither understand nor do spiritual things (1 Cor 2:14; Rom 3:10-11). Yes, God revealed Christ’s identity to Peter (Matt 16:17). Yes, God opened Lydia’s heart (Acts 16:14). Yes, our faith is the work of God (John 6:29). No, regeneration does not precede faith.
No, John 3:16 does not teach that faith precedes regeneration because final salvation not regeneration is in view. No, Acts 16:31 does not teach that faith precedes regeneration because the terms “saved” and “regenerated” are not synonymous. No, Romans 5:1 and Luke 13:3 do not teach that faith precedes regeneration because justification is not the same as regeneration. No, 2 Peter 3:9 does not teach that faith precedes regeneration simply because it doesn’t mention regeneration. Yes, faith does not precede regeneration.
Regeneration is like the wind. We first experience regeneration in faith, but regeneration’s comings and goings are a mystery. While theologians may understandably be curious concerning how regeneration operates, they are not told and cannot know. As Wittgenstein said, “Whereof one cannot speak, thereof one must be silent.”
The wind blows where it wishes, and you hear its sound, but you do not know where it comes from or where it goes. So it is with everyone who is born of the Spirit (John 3:8).
For what it’s worth, we do know that hearing the gospel precedes faith (Rom 10:14).