I was briefly part of a Disciples church in southern Illinois, and studied their theology and history in seminary. It's somewhat difficult to answer some of these questions due to their history. You'll probably have more questions than answers when I'm done.
The Disciples were the original group that stemmed from the Stone/Campbell movement, with a view of believer's baptism that approached baptismal regeneration. Stone and Campbell indeed sought ecumenical unity, but on their terms only (mainly rejection of infant baptism and adherence to their particular means of interpreting scripture). After the Civil War, what is known in the US as Church of Christ split off over a difference of opinion on instrumental music in worship. The DoC had no problem; the Church of Christ's hardline adherence to the regulative principle led them to reject any musical instruments in worship.
They are trinitarian in belief and practice. However, you will find some hesitancy among them about using the actual word "Trinity", as Barton Stone refused to use the term as it was not found in Scripture. They do baptize with the Trinitarian formula.
As for acceptance of infant baptism, it's complicated. Technically, their doctrine rejects it, but there's a bit of a catch. As the years progressed, especially as you get into the 20th century, their ministers were strongly influenced by liberal German theology. While the Churches of Christ and independent Christian Churches (another split after WW2, this one over liberalism) moved to a more exclusive stance, the Disciples leaned strongly into ecumenism. Coming into the 1960s, their desire to be more inclusive led them to become an actual denomination with a centralized hierarchy. As Disciples' membership shrank, they sought out partnerships with other mainstream denominations. They ended up in what they call a "covenant partnership" with the United Church of Christ (not to be confused with the acapella gang earlier in the post). In practice, it's almost a merger in everything but name, with the majority of UCC and DoC ministers being able to serve either denomination.
Here's where things get confused. The United Church of Christ, with their German Reformed and Congregationalist roots, practice infant baptism almost exclusively. The Disciples are credobaptists. So when a family makes the jump from a UCC congregation to a DoC congregation, the question is raised "well, Junior was baptized at the UCC when he was a baby. Does he have to be immersed now?". The solution is generally for the minister to shrug and say "whatever..."
Oh, this is coming back to me now from my UCC days... I remember now. I recall there was some major liberalism in the Disciples in the 60s and to their chagrin they had ordained Rev. Jim Jones, the evil communist cult leader, but later excommunicated him, however, I remember them as the denomination where Ronald Reagan and LBJ were baptized.
By the way, I was always told that the Christian Church was the more formal, liturgical branch, and the Disciples of Christ were less formal and more ex tempore. Was that the case in the 20th century after the Regulative principle of worship was discarded or is it completely backwards?
This is actually thrilling because just as I am injecting Orthodoxy into Congregationalism, I have an idea for a Creedal Christian Church / Disciples of Christ which would accept the idea of the centrality of the Eucharist and ecumenical reconciliation, and of having Elders (which is what Presbyter actually means) chosen by acclamation from the people of the parish as in the early Church consecrate the Eucharist, most of whom would be volunteers, with Superintendents visiting the parishes to ensure doctrinal correctness. The main difference with the Stone/Campbell movement would be that this new grouping, attached to my existing congregational ministry, would regard Stone and Campbell as correct in their emphasis on the Eucharist and desire for ecumenical reconciliation, but in error regarding their anti-creedalism and opposition to infant baptism.
This model would let me plant churches simply called Christian Churches that would embrace core principles of the Stone/Campbell movement that are correct, and which would resemble their worship, with a central Eucharist consecrated by the parish elders and not necessarily the preacher (the Methodist tradition of licensed preachers is also appealing to integrate here) which would avoid their errors, and teach Orthodox doctrines as one finds in Saints Athanasius, John Chrysostom, Cyril of Alexandria, Maximus the Confessor, Severus of Antioch, John of Damascus, John Cassian, Ambrose of Milan, Vincent of Lerins, the Cappadocians (Basil of Caesarea, his brother Gregory of Nyssa and his best friend Gregory Nazianzus) and Gregory Palamas.
Because that was really the main mistake poor Stone and Campbell made. Their heart was really in the right place, and they anticipated the Ecumenical Movement, but they did not know enough about the Early Church or the Eastern churches and the extreme freedom of belief therein. However their idea of a church intended to be anti-schismatic was beautiful and much better executed than the previous attempts at this, such as the Universalists, who were inherently Heterodox, and King’s Chapel in Boston, an Anglican parish which edited the Book of Common Prayer to delete all references to the Trinity, and still uses a version of this butchered* BCP, in order to be able to make welcome anyone regardless of their views on the Trinity; they unsurprisingly joined what is now the Unitarian Universalist Association.
*I say butchered because the doctrine of the Trinity was explicitly suppressed in the King’s Chapel BCP, unlike in the case of the Stone/Campbell movement which like many hesitates to use the word Trinity because it is not in the Bible, indeed, amusingly enough it was coined by Tertullian, a brilliant early theologian who is not venerated as a saint because his rigorism, his belief in the impossibility of the forgiveness of sins committed post-Baptism, led him to join the Montanist cult, whose founder Montanus claimed to be the Paraclete and was followed by three women who were supposedly prophetesses or oracles of some kind.
