Was there unanimous consensus of the Church Fathers in favor of BoD?
I will rightly state all the Fathers who spoke to the issue were on the side of BOD. Others speaking on EENS were not addressing the BOD issue. You wrongly will disagree.
Did Trent teach BoD?
Even Feeney understood that it taught a man can be justified apart from water. I will say Trent taught BOD as Bellarmine, Ligouri, Pius IX and Pius XII show and you will wrongly disagree.
What does it mean that most modern theologians promote BoD?
Here I will rightly claim you have to distinguish between pre and post V2 theologians as most pre-v2 theologians were sound, at least before V2. The serious minded Catholic, for instance, would not dream of disparaging Fenton as you do. The sound theologians referenced in the first half of the 20th century agreed with what had been taught by the Fathers, Doctors, TRENT, Saints, Popes before them. They meant that a non-member who dies in the state of sanctifying grace dies within the Church and is saved. They taught one must die with supernatural Faith and perfect charity to be saved. One guilty of mortal sin, in addition to supernatural Faith must die with perfect contrition in order to be saved. Who dare argue with this? Since when is water more important than Faith!?!
Under what conditions can one receive BoD?
Under the condition of death (while in the state of sanctifying grace).
This is why they are not considered "members" of the Church as we know what the definition of authentic membership is. Those who benefit from BOD become members of the Church Suffering or Triumphant at the moment of death, though they are within the Church (despite not being members) by desire. They are in a state of sanctifying grace which can only be obtained within the Church. One does not desire that which one already has. One who has been baptized does not benefit from being baptized again or from desiring to be baptized. This is incredibly basic stuff.
They must be inculpably ignorant of the necessity of Baptism and of the need to be attached to the Church. Anyone who is aware of the necessity of baptism and refuses or needlessly puts it off will be damned. The same holds in regards to joining the Church. Anyone who refuses to do this cannot have perfect charity and be in a state of sanctifying grace.
Is explicit faith in the Holy Trinity and Incarnation necessary for supernatural faith and salvation?
I have answered this question many times in great detail and it has been ignored. I may post it AGAIN if I get a chance. When addressing the issue right through Pius XII they did not specify with what necessity, if any, explicit faith is necessary in the Incarnation and Holy Trinity. The issue was addressed under Pius XII and by Pius XII. It was addressed by Pius IX, by Aquinas, by the theologians and they did not settle this issue in regards to BOD. I have quoted Aquinas on this several times.
As a side note your tactic of using authorities when it suits your purpose and disparaging them when they clearly disagree with what you prefer to believe is a bit off-putting. That is about as charitable as I can put it.
What is specified clearly is the need for supernatural Faith which clearly is the belief in God who rewards and punishes at a minimum.
They speak of supernatural Faith, and list belief in God and repentance as being necessary for salvation to be possible. They go no further neither affirming or denying the explicit Faith in the Holy Trinity and Incarnation are necessary with an intrinsic necessity of means under every and all circuмstances. I can accept this because I accept whatever the Church teaches and accept as unsettled whatever has not been settled. Why do you think you know more than the experts?
All the traditional clergy, who are more qualified than us, whether R & R or SV agree. And if they can all agree on something as controversial as this it should give us pause. Fenton who studied the issue in depth and who studied what his fellow theologians taught acknowledged that the theologians were divided on the issue as late as 1958. If it was not settled then it is not settled now. Why do you raise yourself above this people as the sole rule of faith?