What was the rationale behind the various Church Councils choosing certain books to be included in the *first* instance of the Bible as canonical, and not others?

(I've missed a few of your direct questions, I can come back and clarify them if the below doesn't make it clear enough.)

It’s a common misconception that the New Testament Canon (i.e. the idea of and collection of a set of books that was authoritative for the church) was ‘decided’ at some ‘point’, usually by a council. The most common ill-conceived date for this is the first Ecumenical council, Nicaea 325, which was called by Emperor Constantine I. Actually, this council did not discuss the canon at all.

Actually, the development of the NT Canon was primarily a process, which latter appears to be ratified by several councils.

Obviously, in the Early Church (late 1st century) there are no nicely stitched together ‘New Testaments’ circulating, instead there are individual documents written to individuals or more commonly churches, which are treated with some degree of authority and prestige. These documents would have been read aloud in public settings, copied and transmitted for other groups of believers to make use of, and over time collated.

By the time of the second and third generations, i.e. those called ‘Apostolic Fathers’ (because they lived while the Apostles were alive, rather than that they were Apostles themselves), you get treatment of some of these texts as authoritative, i.e. by referring and quoting to them. You see this, for example, in Polycarp’s letter to the Philippians, in 2 Clement (2.4), in the Letter of Barnabas (4.14). Justin Martyr refers to the ‘memoirs’ of the apostles and records that they were read in Church along with the readings of the Prophets (i.e. OT writings).

One clear, though disputed, impetus for the process of canonisation was the role of Marcion. Marcion was active around 130-140 and himself gathered a collection of scriptures into an authoritative ‘canon’, which included a Gospel (an edited version of Luke) and a collection (also edited) of Pauline epistles terms the ‘Apostolikon’. Marcion’s theology relied on a firm disjunction between an OT God and the NT God and has some features of dualistic gnosticisms. Overall Marcion and his views were firmly rejected by the majority streams of the churches, but it also highlighted the question of texts and authority.

Around the same time you see a proliferation of (other) Gnostic texts which pushes the churches to clarify which texts they treat as authoritative and which as not, and why.

Once you get to someone like Irenaeus (in the second half of the 2nd century), you get in his writings a wide acquaintance with and citations from most of the texts that compose the New Testament. It’s clear that those texts have wide geographic dispersion and function authoritatively in various Christian (proto-orthodox) communities. The earliest text that we have that contains a ‘canon’ list is the Muratorian canon, usually dated to the 2nd century. This text mentions 4 gospels, but names Luke and John (the first two are missing from the fragment), Acts, 13 Pauline epistles, 2 letters by John (probably 1 and 2 John, but maybe 2 and 3?), Jude, and Revelation. It omits Hebrews, 1 and 2 Peter, and a third Johannine letter. It also accepts Wisdom of Solomon, an Apocalypse of Peter (though it’s unclear which text this refers to, there are 2 possibilities), though mentioning that some do not permit its public reading in church. Similarly Shepherd of Hermas is accepted but not to be read publicly. It then lists several groups and texts that are to be outright rejected.

The important early 3rd century theologian Origen uses the full range of the NT canon, though expresses some doubts about James, 2 Peter, and 2 and 3 John, while at other times utilising with approval the Gospel of Peter, G of the Hebrews, Acts of Paul, 1 Clement, Epistle of Barnabas, Didache, and Shepherd of Hermas; he denies authenticity to Gospels of Thomas, of the Twelve, of Basilides, of the Egyptians, of Matthias, and Preaching of Peter.

A list that is really beginning to look like the modern NT appears in Eusebius of Caesarea, Eccl His 3.25.1-7 and also 3.3.5-7, in which he lists four categories of books: recognised, disputed, spurious, and heretical. The main issues are that he lists Hebrews as both accepted and disputed, James, 2 Peter, 2 and 3 John, Jude as disputed, and Revelation as both accepted and possibly spurious. He doesn’t include any text as recognised that isn’t in the standard 27 book canon. The first list that clearly matches with zero ambiguity the present 27 book NT Canon is Athanasius’ 39th Festal letter, dated to 367 AD. He also lists books not placed in the canon, but suitable for reading and instruction in piety, as well as other books that are fabrications.

Mostly after this do you get councils that discuss canon lists, including the Synod of Laodicea in 363 (note that this precedes Athanasius’ list), which discusses the issue, but Canon 60 that lists the books (all of the NT except Revelation) is likely a later addition. The Councils of Carthage in 397 and 419 summarise the acts of a Synod of Hippo Regius (393) that accepted a 26 book canon, to which Revelation was officially added in 419. This shapes the Western canon. Around the same time a consensus appears to be emerging in the West, as at the Council of Rome in 382, which shapes the Vulgate translation.

Things develop differently in the East in late antiquity, especially outside the Greek speaking churches; Revelation was often suspect, and in some Eastern churches is not normally read in churches . The Church of the East using the Peshitta has a 22 book NT; there are other variations in Eastern churches.

So, in short, not a conciliar decision. A process of recognition and acceptance and formulation of what was widely and authoritatively accepted.