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So, I'm going to address two issues mainly in this post:

1) The use of 1Co 2:14 as mistaken proof that the unregenerate can not believe the gospel.
2) The use of Acts 10 as mistaken proof that the unregenerate can not believe the gospel.

It will be shown both turn out to actually be arguments from silence.

(All cites are NASB)

1) The use of 1Co 2:14 as mistaken proof that the unregenerate can not believe the gospel.

Starwind post #265:

1Co 2:14 But a natural man does not accept the things of the Spirit of God, for they are foolishness to him; and he cannot understand them, because they are spiritually appraised.
I believe when Paul wrote things of the Spirit of God ... they are spiritually discerned he meant things like gifts of the Spirit, fruit of the Spirit, as well as "THINGS WHICH EYE HAS NOT SEEN AND EAR HAS NOT HEARD, AND which HAVE NOT ENTERED THE HEART OF MAN, ALL THAT GOD HAS PREPARED FOR THOSE WHO LOVE HIM." - things of the kingdom of heaven.

That’s not what the verse says.

Alrightythen. From the top, here is the context of what 1Co 2:14 is about, and to what 'things of the Spirit of God that natural man can not understand' Paul refers.

Divisions had arisen in the church at Corinth and Paul was correcting them in 1Co 1:10-4:21
Paul steps through the causes of those divisions in 1Co 1:10-2:16
Paul specifically addresses misunderstanding of the Spirit's ministry of revealing in 1Co 2:6-16
Paul further addresses the consequences of those divisions in 1Co 3:1-4:5

Picking up now with Paul addressing the misunderstanding of the Spirit's ministry of revealing in 1Co 2:6-16:

1Co 2:6-9 Yet we do speak wisdom among those who are mature; a wisdom, however, not of this age nor of the rulers of this age, who are passing away; but we speak God's wisdom in a mystery, the hidden wisdom which God predestined before the ages to our glory; the wisdom which none of the rulers of this age has understood; for if they had understood it they would not have crucified the Lord of glory; but just as it is written, "THINGS WHICH EYE HAS NOT SEEN AND EAR HAS NOT HEARD, AND which HAVE NOT ENTERED THE HEART OF MAN, ALL THAT GOD HAS PREPARED FOR THOSE WHO LOVE HIM."

Paul begins addressing their misunderstanding of the Holy Spirit by pointing out that hidden wisdom predestined by God which no rulers of the age understood is spoken to those who are mature.

1Co 2:10-13 For to us God revealed them [wisdom, things] through the Spirit; for the Spirit searches all things, even the depths of God. For who among men knows the thoughts of a man except the spirit of the man which is in him? Even so the thoughts of God no one knows except the Spirit of God. Now we have received, not the spirit of the world, but the Spirit who is from God, so that we may know the things freely given to us by God, which things we also speak, not in words taught by human wisdom, but in those taught by the Spirit, combining spiritual thoughts with spiritual words.

Paul continues in 1Co 2:10-13 connecting his terminology from 'hidden wisdom' to 'things of the Spirit'; teaching how these 'things of the Spirit' were revealed by God to "us" (meaning Paul and those who are mature and presumably including the Apostles); things spoken of using spiritual words combined with spiritual thoughts.

1Co 2:14 But a natural man does not accept the things of the Spirit of God, for they are foolishness to him; and he cannot understand them, because they are spiritually appraised.

And we have arrived at 1Co 2:14 but now within the context that the "things of the Spirit of God... spiritually appraised" is the aforementioned 'hidden wisdom spoken to the mature'.
Paul is teaching that the 'hidden wisdom - things of the Spirit of God' spoken of with spiritual thoughts and words, are 'spiritually appraised', and 'not understood by natural man'.

1Co 3:1-3 And I, brethren, could not speak to you as to spiritual men, but as to men of flesh, as to infants in Christ. I gave you milk to drink, not solid food; for you were not yet able to receive it. Indeed, even now you are not yet able, for you are still fleshly. For since there is jealousy and strife among you, are you not fleshly, and are you not walking like mere men?

In 1Co 3:1-3 Paul, refering back to his initial teaching at Corinth (on his 2nd journey), describes their need as unspiritual men for "milk" - the gospel, rather than the "solid food" suitable for spiritual men.
Paul says that the 'hidden wisdom things of the Spirit of God' that are 'spiritually appraised' are "solid food" and *could not* be spoken to those at Corinth because they were (and still are) fleshly and instead needed to hear "milk". What, if not the gospel, would Paul have given the unregenerate at Corinth as milk? Paul makes clear that the "solid food" of things of the Spirit can not be given to men of flesh but instead they need "milk"

1Co 3:6 I planted, Apollos watered, but God was causing the growth.

Paul "planted". What would Paul plant other than the seed of the gospel? The gospel is not what Paul meant in 1Co 2:14 as "things of the Spirit" or in 1Co 3:2 as "solid food". The gospel is the seed Paul planted; the "milk" needed by unregenerate fleshly men to become infants in Christ.

So Paul makes a distinction that the things of the Spirit of God of 1Co 2:14 are the "solid food" intended for mature spiritual men, whereas the gospel is the "milk" (not mentioned in 1Co 2:14) as suitable for fleshly men and infants in Christ.

And so to argue that 1Co 2:14 teaches that the unregenerate can not believe the gospel ("spiritual milk") if and when heard is to argue from silence, because 1Co 2:14 was:

not addressed to the unregenerate needing "milk" (the gospel), but rather was;
addressed only to the mature needing "solid food" (hidden wisdom things of the Spirit of 1Co 2:6-9 spoken to the mature).

2) The use of Acts 10 as mistaken proof that the unregenerate can not believe the gospel.

I believe your case of “faith comes from hearing the word of God” falls apart under the example of Cornelius in scriptures.

You'd best take this up with Paul then (Rom 10:17 So faith comes from hearing, and hearing by the word of Christ).
Your quarrel is really with him, as I merely relied on his statements. Charitably, how could you post near verbatim scripture and say it falls apart under some other scripture? Is not all scripture inerrant and self-consistent?
So, what does Acts 10 record, actually? I won't repeat it all, but here are the pertinent highlights:

Act 10:2 [Cornelius - a gentile] a devout man and one who feared God with all his household, and gave many alms to the Jewish people and prayed to God continually.
Act 10:3-5 was visited by an angel who said Cornelius' "prayers had ascended as a memorial to God" and verbally directed Cornelius to send for Peter.
Act 10:9-20 God gives Peter a vision that (ostensibly what Gentiles eat) is not to be considered unholy but cleansed, and that Cornelius has sent for him.
Act 10:22 Cornelius is described as a righteous and God-fearing man well spoken of by the entire nation of the Jews.
Act 10:28 Peter says God has shown him (Peter) to call no man (ostensibly Gentiles) unholy or unclean.
Act 10:33 Cornelius and his household are assembled to hear what Peter says.
Act 10:37-38 you yourselves know the thing which took place throughout all Judea, starting from Galilee, after the baptism which John proclaimed. "You know of Jesus of Nazareth, how God anointed Him with the Holy Spirit and with power, and how He went about doing good and healing all who were oppressed by the devil, for God was with Him.
Act 10:44-45 While Peter was still speaking these words, the Holy Spirit fell upon all those who were listening to the message. All the circumcised believers who came with Peter were amazed, because the gift of the Holy Spirit had been poured out on the Gentiles also.

So, Cornelius (a Gentile) knew before Peter's visit (and before the Holy Spirit fell) of Jesus ministry, miracles, and possibly teachings, and Cornelius had been a devout God-fearing man whose prayers were a memorial to God. Further, God considered Cornelius worthy of an angelic visitation with a verbal command that was to initiate, via Peter, the Gospel coming to the Gentiles.
Clearly, Cornelius believed in God and knew of Jesus' ministry long before Peter showed up. Cornelius already believed; believed in what precisely we don't entirely know, but it is entirely reasonable for us to understand from Act 10:38 that Cornelius minimally knew of and believed that Jesus was the anointed Son of God and Messiah. What else Cornelius first heard from Peter about Jesus' teachings is not clear. What is clear is that Cornelius previously 'heard' of and 'knew' (believed?) Jesus was the Son of God. After Peter taught further, then the Holy Spirit fell on all of them assembled, not just Cornelius.
There is nothing in Acts 10 that demonstrates Cornelius' regeneration (salvation and sealing) prior to belief and belief prior to hearing.
One must argue from silence and assume (in support of ones doctrine) that Cornelius was first regenerated and then believed and then heard - and that at this point Acts 10:1 picks up the story. In fact, arguably, regeneration, renewing and sealing occurred when the Holy Spirit fell - if one were to overreach.
But again, my simple point is that nothing in scripture supports OP's contention that (in OP's words):

it is impossible that a man should believe and trust the Gospel while he is Unregenerate, then -- In order to believe and trust the Gospel, is it necessary that an unsaved man must first be Regenerated by the Holy Spirit

Consequently, there is nothing about Cornelius (Peter, yes) in Acts 10 that demonstrates:

1Co 2:14: But a natural man does not accept the things of the Spirit of God, for they are foolishness to him; and he cannot understand them, because they are spiritually appraised

Again, Paul in 1Co 2:14 is referring to spiritual "solid food", not the "milk" suitable to men of flesh - infants in Christ.

Cornelius, prior to Peter's arrival, was already a devout God-fearing prayerful man who knew of and seemingly believed Jesus was the anointed Son of God. Cornelius knew these things by report; he had heard about them, perhaps had even read OT scripture, and he believed what he'd heard. What Cornelius heard and believed was not "solid food"; were not "things of the Spirit of God" of which Paul was teaching that could only be "spiritually appraised".
No, Cornelius had heard and believed "milk"; about the Israelite God and Jesus' miracles - things that would comprise the news headlines and watercooler discussions of the day, and Cornelius believed based on what he knew and the evidence heard, that the Israelite God existed and Jesus was His miracle-performing Son. That is not an insurmountable spiritual leap. That is simply believing what the evidence shows.
Acts 10 demonstrates spiritual "milk" had been consumed by Cornelius - an infant in Christ if not a fleshly man.
To otherwise argue that Cornelius was spiritually mature (in the sense conveyed by 1Cor 2) and that his fearing God and believeing Jesus ministry constitutes spiritual "solid food" is to argue from silence again, as Acts 10:37-38 bareley describes the gospel ("milk") let alone the hidden wisdom spiritual things of 1Co 2. Further, Paul's "solid food" teachings are considerably deeper than fearing God and believing Jesus.

Note carefully that I'm not changing my argument that the spiritual things of 1Co 2:14 are "solid food", whereas what Cornelius heard and believed about Jesus is what Paul termed "milk" suitable for men of flesh who could not be spoken to as spiritual men. (1Co 3:1).

Keep in mind your two arguments with 1Co 2:14 and Acts 10 were arguments from silence.

Addressing now some of your closing comments:

Your error is in boiling down the gospel to just simply speaking, hearing and making some type of “intellectual” choice. There is no room in your interpretation for the working of the Holy Spirit. In fact, I haven’t seen that mentioned very much at all.

Again you are trying to recast my argument that many passages of scripture show, simply, that belief in the gospel follows hearing the gospel and precedes regeneration (saved and sealed). No more, no less. But that is an argument (and its accompanying scripture) you keep deflecting to some other argument you'd rather have. I dwelt primarily on scripture showing man's ability to believe or disbelieve the gospel after hearing it. The listening (or not) and believing (or not) are done by man - not the Holy Spirit.

You seemingly want to see an argument wherein the Holy Spirit first does the regenerating and then does the believing on behalf of the "listener" and only then they listen to the gospel. So, no, I didn't mention the Holy Spirit in that capacity as I don't find it supported in scripture. But I did point out the Holy Spirit does the regenerating and sealing. Nor did I mention Moses, nor the Trinity, nor the crucifixion, nor the rapture, nor ... well if you expected a condensed bible in one post, I'm sorry to have disappointed you.

I stated quite clearly what my focus was (that belief in the gospel follows hearing the gospel and precedes regeneration), and I gather I was successful as your argument seems to be mainly with what you wanted me to say rather than what I did say.