As regards your question about “baptismal regeneration”: It depends on what you mean by the phrase. If you mean the Roman Catholic doctrine which holds that an infant is automatically regenerated in baptism, then no, we don’t believe in baptismal regeneration. However, we do believe that Trinitarian baptism does bring people into covenant with Christ. Baptism is spoken of in Scripture as uniting people to the death and resurrection of Christ (Rom. 6:3-4, Col. 2:11-12), and this union is regeneration IF it is received by faith (note Col. 2:12) but it is not automatic. As we understand it then, baptism does truly bring about a covenant relationship between the person baptized and the Lord Jesus. But we are saved by grace through faith (Eph. 2:8). All that to say, we would describe baptism as something like a wedding. There is a real covenantal union that takes place, but covenants are kept by faith or broken by unfaithfulness. Faithfulness/unfaithfulness will fulfill that covenantal union (salvation) or destroy it (apostasy/damnation). And of course as Calvinists, we believe that all of this is ultimately in the hands of our heavenly Father, but this does not negate our own human responsibility to perservere in faith, clinging to Jesus Christ. We should also note that “faith” is something that is often attributed to little children and infants; Jesus even makes their faith a model for us. So while we still do not want to insist on any sort of automatic baptismal regeration, a more full-orbed biblical approach would nevertheless see baptismal regeneration as far from impossible and perhaps more frequent than we commonly think, not because of some kind of magic in the water, but given what Scripture says about infants and little children and their faith (e.g. Ps. 22:9-10, Mt. 18:3, Mk. 9:42, 10:15).
Of course Scripture is our ultimate authority, but the Westminister Confession of Faith is our church’s secondary confessional statement. As you’re probably aware, this statement is one of the most widely accepted confessions of faith for historic Presbyterians.
Below is the WCF statement on baptism. Notice the first paragraph that describes “regeneration” as one of the things that baptism is a “sign and seal” of, but also notice the fifth and sixth paragraphs which make it clear that salvation is not inextricably tied to baptism and that the grace of baptism may be applied by the Holy Spirit in varying ways and at various times.
I. Baptism is a sacrament of the New Testament, ordained by Jesus Christ, not only for the solemn admission of the party baptized into the visible Church, but also to be unto him a sign and seal of the covenant of grace, or his ingrafting into Christ, of regeneration, of remission of sins, and of his giving up unto God, through Jesus
Christ, to walk in newness of life: which sacrament is, by Christ’s own appointment, to be continued in his Church until the end of the world.
II. The outward element to be used in the sacrament is water, wherewith the party is to be baptized in the name of the Father, and of the Son, and of the Holy Ghost, by a minister of the gospel, lawfully called thereunto.
III. Dipping of the person into the water is not necessary; but baptism is rightly administered by pouring or sprinkling water upon the person.
IV. Not only those that do actually profess faith in and obedience unto Christ, but also the infants of one or both believing parents are to be baptized.
V. Although it be a great sin to contemn or neglect this ordinance, yet grace and salvation are not so inseparably annexed unto it as that no person can be regenerated or saved without it, or that all that are baptized are undoubtedly regenerated.
VI. The efficacy of baptism is not tied to that moment of time wherein it is administered; yet, notwithstanding, by the right use of this ordinance, the grace promised is not only offered, but really exhibited and conferred by the Holy Ghost, to such (whether of age or infants) as that grace belongeth unto, according to the counsel of God’s own will, in his appointed time.
VII. The sacrament of Baptism is but once to be administered to any person.