Dean:
1 Tim. 2:5
I think it's "Godhead" in both cases. Christ is the frontman, so to speak, representing the collective.
Ambassador / Mediator
I wasn't confusing the roles, because this IS a special case.
In this case the human race IS estranged from that of "God".
The Ambassador has come from the aggrieved party, to mediate in the form of BOTH natures (God AND Man), because it appears there is no other third party in a position to do this.
I would offer you Gal.3:20, by way of support of this.
Begotten EternallyMaybe the word eternal means something different to you, than it means to me, in the context of "God". To me it means "outside of time" (not "infinite time" - that's another concept).
I believe that if it were possible to state that the Son had a beginning, then it was before time even existed. If there was such a beginning, it would have been at this point that the Father became "the Father". Before such an event and after it there only existed "God".
Finally creation began, including that of time, through the Son.
Father Begotten?
That one is faintly ludicrous - LOL
A father is a father, however only becomes such upon the generation of a child.
Also the Father is not the Son. Of course there is a difference between the existence of each. I see no argument for declaring them symbiotic, either, given that one of them died.
Amoeba
In the case of the single-celled organism, a split occurs, with each portion retaining an identical half, before they go on to lead individual lives, whilst still remaining of the genus "Amoeba".
I don't think you'd find it particularly easy to differentiate which cell was the older.
Again I tell you that the Trinity doctrine, as I understand it, doesn't say that the Father IS the Son. It openly acknowledges that these are two distinct persons.
Pre-dating
To "pre-date" something, you need time to exist.
This is my view of eternity - not infinite time, but an abstract in it's own right in which God dwells.
I hope that helps you understand my position a little better.
As with all things to do with the substance of "God", it's beyond our ken.
Therefore it is the role of the philosopher to try to make it understandable for human understanding.
I believe we will never truly "comprehend" it, however.