Good questions! I think that there are a number of issues at play in this discussion. I'll give you some thoughts that you can chew on and ask some more. I would heartily recommend Doug Wilson's book "To a Thousand Generations." I found it particularly helpful as I wrestled with these issues. The difference between credobaptism and paedobaptism is like two different sets of prescription glasses. Hence, it is challenging to isolate the real differences between the two in short space. There are lots of intertwined issues and it has taken me years to work through them - indeed, I'm still working! But let me try to hit a couple major points - I may not hit all your questions so ask again if I miss something that is important.
Two central, related issues in this debate are the nature of the new covenant and the meaning of baptism. On the one hand, credobaptists insist that the new covenant includes only believers ("all shall know me, from the least to the greatest" - Jer 31:34). Because credobaptists insist that the new covenant includes only believers, they thereby endeavor to limit baptism to those who have made a personal profession of faith and thus given personal evidence of regeneration. While this evidence is not absolute (witness the case of Simon the magician in Acts 8), this evidence at least gives us more confidence that the individual is personally converted than we would have otherwise. Baptism, in this view, is an evidence of the individual's faith, an external evidence of an internal change.
Paedobaptists, on the other hand, argue that the new covenant includes believers and unbelievers. There are branches "in Jesus" that do not bear fruit and must be pruned (Jn 15:1ff). There are those who have "become partakers of the Holy Spirit" who fall away (Heb 6:4ff). There are those in the new covenant who "trampled the Son of God under foot, and counted the blood the covenant by which they were sanctified a common thing, and insulted the Spirit of grace" (Heb 10:29). Arminians insist that such passages teach that we can lose our salvation, that there is no such thing as the perseverance / preservation of the saints. But we know that's not the case. Jesus promises that He will lose none of those who are given to Him (Jn 6:39). So Reformed paedobaptists argue that these passages refer not to the loss of individual salvation, as though God's individually elect could perish, but to the loss of covenant status and identity. Those who fall away were corporately elect but not individually elect. "Not all Israel is Israel." But, and this is a critical point, all Israel should be Israel. Having been marked out by God as His own with the sign of the covenant, they should reflect that identity in their hearts (Dt 10:16; 30:6). Circumcision marked them out as God's people in the old covenant and baptism, in the new covenant, so marks us.
Reformed credobaptists end up, in my opinion, having to explain these warning passages away - they are hypothetical warnings; the people may have been members of the visible church but not of the new covenant; some such rationale is used. However, Hebrews is the book that develops Jeremiah's promise of the new covenant (Heb 8) while simultaneously warning those in covenant with God not to fall away (Heb 2, 6, 10). So what this means, I think, is that the new covenant includes both genuine believers (those who fully partake of the meaning of the new covenant) and false believers (those who are members of the covenant but not in a living sense). So I would argue that Judas was a "Christian" in this sense as was Simon the magician. They both were members of Christ (Jn 15) but not in a living fashion. But precisely because they were members of the new covenant, they were more culpable for their unbelief rather than less (Heb 2:1-4).
Consider the parallel of an unfaithful husband. We can talk about that husband in a couple different ways. Is he a husband? Yes, absolutely! That's why he is called an adulterer and not a fornicator. But, on another level, we can ask the question, "Is he a husband?", and answer with a resounding, "No!" He is not being faithful to his wife, he is not being what a husband ought to be. But precisely because he is a husband he is culpable for not being a husband! It is his covenant with his wife that makes him doubly guilty - guilty of sexual sin and guilty of covenantal unfaithfulness.
So in the paedobaptist understanding, baptism makes us members of the new covenant, unites us to Christ covenantally, and summons us to a life of faithfulness and discipleship. Baptism is "a sign and seal of the righteousness we have by faith" (cf. Rom 4:11). Note, therefore, that it is not a sign of our faith - it is a sign of the righteousness we have by faith. And what righteousness do we have? Is our faith meritorious? Do we have a personal righteousness to which baptism points? No! Absolutely not! Baptism doesn't point inward to me and my faith but outward to Jesus and His righteousness - He is the righteousness that I have by faith. Baptism is God's Word to me, promising that all those who trust in Jesus for righteousness, forgiveness, and salvation will in fact be delivered from their sin.
Baptism corresponds, therefore, to the "vow" that a husband and wife exchange. In the case of baptism, it is God's vow, God's promise to be our God. On our side, it marks us out as God's child, separate from the world and devoted to Him, "saints."
Credobaptists, in my opinion, end up drawing distinctions between the OT & NT people of God that the NT doesn't draw. Paul warns the Corinthians to not be like our fathers in the OT; this seems to presume that it is possible for us to become like them. So Paul says that the Corinthians, like our fathers, have received baptism and the Lord's Supper (1 Cor 10:1-2) but this is no guarantee of God's smile - after all our fathers were "baptized" and "ate the same spiritual food and drank the same spiritual drink (Christ)" and yet died in the wilderness - they fell under the judgment of God. Paul's words imply that there are members of the new covenant who likewise fall under the judgment of God.
So how are we to understand the promise that "all shall know me, from the least to the greatest"? Personally, I think that that promise is eschatological - it looks forward to the eventual spread of the Gospel throughout the nations of the earth. God's promise is that every knee shall bow and every tongue confess that Jesus is Lord - some in judgment and some in salvation. Jeremiah's promise implies that the number of the saved shall be massive, myriads upon myriads. In addition, his promise insists that when God pours out His Spirit, there is a universal knowledge of God among His people - God preserves us from men like Judas and Simon.
However, in the course of history, there are often tares among the wheat; there are folks who fall away in times of persecution or who are overcome by the lust of the flesh and the desire for things of this life (parable of the Sower). These folks were members of the church and of the new covenant (consider Jesus' words to the seven churches in Revelation 2-3) who turned away from God and incurred His judgment. They went out from us because they were not of us, for if they had been of us they would have remained with us (1 Jn 2:20).
So these are two of the "watershed" issues that separate credobaptists and paedobaptists.
I certainly grant that there are distinctions between the old covenant and the new covenant. But these distinctions are chiefly of the "new covenant has more" variety. Old Covenant = Gospel primarily in Israel; New Covenant = Gospel to all nations. Old Covenant = Sign applied only to men; New Covenant = Sign for all members. Old Covenant = Ethnic Israel; New Covenant = Spiritual Israel. But even in the OC there were hints and anticipations of some of these things - Rahab, Ruth, Nineveh, Psalms, etc. So to address whether children are viewed differently in the new covenant, we'd have to ask what the NT teaches about kids (and also what the OT prophets taught about kids in their prophecies). And what we find is glorious continuity - Jesus blesses the children, even infants, of his disciples (Lk 18:15ff); Paul issues his commands to "households" which includes kids and he exhorts the kids to obey their parents "in the Lord" (Eph 6). So there is continuity in the way we are to view our children - they are members with us of the kingdom of God and are to be brought up in the faith to love and cherish the ways of the Lord. By nature they, like we, are "outsiders" and "children of wrath"; but, by grace, they are incorporated into the people of God and marked out as God's own children, summoned to walk with Him all their days.