1) “If sin is the result of law as you say…”

Not as I say, as Paul says, this is plain statement scripture:

Rom 4:15
Because the law worketh wrath: for where no law is, there is no transgression.

1a) “then how can 3:23 be true, given that not all have heard the law?”
Rom 1:18-20
For the wrath of God is revealed from heaven against all ungodliness and unrighteousness of men, who hold the truth in unrighteousness;

Because that which may be known of God is manifest in them; for God hath shewed it unto them.

For the invisible things of him from the creation of the world are clearly seen, being understood by the things that are made, even his eternal power and Godhead; so that they are without excuse:

2) How can Paul write to Gentiles that they were dead in their sins and trespasses (Ephesians 2:1-3 – note particularly verse 3, “were BY NATURE children of wrath”; Colossians 2:13) if they had not heard the law?
(See response to 1a)

3) Your interpretation also provides no explanation for how believers, generated and indwelt by the Spirit, can continually sin even after conversion?
Yes it does,
a) If we do not follow the Spirit of God, we are in sin.
b) if we live according to the Law, rather than by the Spirit of God we fall back into sin.

What condition causes the flesh to war against the Spirit in a believer (Galatians 5:17)?

4) Finally, what would be the cause of the “spiritually alive” and Spirit-indwelt Apostle Paul’s description of indwelling sin in Romans 7:17,20? What is this “law” that causes him to continually do evil (Romans 7:14-23) even after becoming “spiritually alive”?
Great question! So glad you asked this!
Romans 7 is about the Law of Moses, it has been bastardized it to mean something else, but if you follow the context of Romans, Paul is speaking of the way of salvation to Jews and Gentiles. Romans 7 is entirely about what he also calls in other places: ‘The curse of the law.’

Paul is clearly speaking of following the Law, vs following the Spirit of God.