I am understanding your use of the word prevenient grace to be not only God's influence upon men before actual justification but also joined with the assumption that his intended influence upon men to draw them to salvation by the Holy Spirit can be rejected and is resistible. In this sense the grace you imagine is something opposed to Calvinism.
To answer the question from the standpoint of a Calvinistic view we must first make clear that Calvinism does not reject the idea that there is a gracious influence from God before actual grace is received through the gospel, rather it upholds it. Calvinism only rejects the idea that it is resistible. The original reason for rejecting the idea of the original Roman Catholic teaching of prevenient grace is because under Catholic theology it is actually a means to introduce the idea that grace is only half grace (from the protestant concept) and always depends on cooperative human works. Under this scheme grace can be earned by cooperating with previous grace and justification is not in a moment but through a lifetime of 'cooperative' works. Therefore, under a reformed evangelical view, Catholic theology actually says that salvation is by human effort 'aided by the Holy Spirit' and is therefore actually just salvation by works. You seem unaware as you toy with the phrase prevenient grace that many theologians see such a devilish background lurking around it.
One can see this line of reasoning from Calvin himself.
the Papists confess that we are under the tyranny of the devil, and slaves to him, and that there is no right will in men, except through the prevenient (præeunte) grace of the Holy Spirit. But as I have already said, they talk vainly of the grace of the Spirit, and say that it is an influence by which God enables us to follow that which is right, if we have a will to do so. Thus, then, the grace of God, according to them, leaves men in suspense, so that they are free either to receive or to reject the grace of God. Afterwards, they join the subsequent grace, which, in their view, is a reward; for if I assent to God, that is, if I suffer myself to be ruled by his Spirit, and embrace the grace offered to me, God will then reward me with another grace to confirm me in my right purpose. And thus they confess that perseverance is in part the gift of God; but they always imagine it a co-operating grace. And then, as perseverance, according to them, is God’s subsequent grace, and is, as it were, a handmaid, it ceases to be grace, for it is rather the reward of merit. (BOOK OF THE PROPHET JEREMIAH and THE LAMENTATIONS BY JOHN CALVIN, p4.213)
However I understand that under your thinking you are associating the idea of prevenient grace NOT as a sinister means to destroy the protestant view of grace. You have a more happy idea in mind, for you seem to be considering people like Baptists or any evangelical that does not believe in the non-resistible grace of reformed theology.
I assume these question arise not from the subject of God's influence on men prior to new birth but are an outcome of wrestling with the idea of election. I can't imagine a big interest in the term otherwise. From this standpoint you may be contemplating a 'limited acceptance' of the concept of 'prevenient grace' to be a possible solution to the unresolvable tension between the idea of free will and predestination. However, I do not think this will really resolve that tension for it does not address the issue before time began but merely in the experience of men with the Holy Spirit in time. This means regardless of how we color a limited acceptance of some kind of grace that can be 'rejected or accepted' it can not prove if it is ultimately resistible or not. For example, this simple Calvinistic type question removes the relevance of prevenient grace from the subject of predestination: 'What if God choses to provide limited influence of goodness towards those destined to hell, only to establish more guilt on their soul for rejecting his prevenient grace, knowing that they would reject it. On the other hand for the elect he provides 'more' prevenient grace which becomes 'irresistible' in order to save those he choses to save?'
I do not actually think God works in a way that we can fully comprehend and understand to relive the tension of election and free will, so my question above is actually not my question. My personal theology is Calvinistic with some associated mysticism, a.k.a..humble admission of not knowing everything. What I mean is all the elect will be saved and actual saving grace is irresistible but I also believe God desires the salvation of all men. The knowledge that can relieve this apparent contradiction is not in my mind declared in scripture, making faith our only refuge and comfort for the desired removal of the seeming tension.
Anyway to answer your specific examples:
Cornelius coming to the Peter even though apparently not yet a Christian in Acts 10
Luke 19:10 This is not an example that opposes irresistible grace as Cornelius eventually becomes a Christian, which rather supports Calvinistic beliefs in irresistible grace
Titus 2:11: "For the grace of God that brings salvation has appeared to all men." This has no bearing on Calvinistic doctrine concerning irresistible grace for of course the grace of God has appeared to all men but only those who are irresistibly drawn to it can take hold of it through faith.
John 6:44: "No man can come unto me, unless the Father who hath sent me, draw him..." This is a actually a proof text for irresistible grace which assume a prevenient and irresistible saving work of the Holy Spirit upon a sinner who is unable to draw himself or properly cooperate in a meritorious way into his salvation.
John 16:18 "And He, when He comes, will convict the world concerning sin and righteousness and judgment;"Again this has no bearing on Calvinistic views concerning irresistible grace, for being convicted of sin and fearing the judgment as a guilty sinner, or being convicted of one's justification by freely accepting the irresistible grace of God is entirely in accordance with Calvinistic views of the work of the Holy Spirit, condemning the damned and saving the elect.
Possibly your only confusion is that you did not realize every Protestant believes in a kind of external grace before actual justifying grace (call it prevenient or whatever). The only difference between some Baptists for example and reformed parts of the church is that this prevenient grace is not resistible with respect to the elect and their being drawn into actual justifying grace.