[Ripe]

I’m trying to better understand the relationship between the Zhenterim and the Drow. IS there one? I would think that the Black network would be interested in at least trade with the underdark. Anyone know the story?

As far as drow of the Underdark are concerned, Black Network would be interested in trade with the drow, but that trade would be conducted in special places like Mantol-Derith, not directly with Underdark drow cities.
As far as drow of House Jaelre or Azukovyn Clan that currently reside in Cormanthor, that relationship could be anything that your DM decides. There is no evidence of that conection in FRCS, but we are still awaiting for book detailing Dalelands, Cormyr and Moonsea region.

Also with the drow, and the churches of Bane, Cyric, and Xim(?). I would like to know how they might interact.

Well, in any of the Lolth controled drow cities it is a crime punishable by death to worship any deity but Lolth (even if I do not belive that extend to people worshiping Selvetarn). In reality that rule is enforced only as far as drow are concerned, other races are (to a point) allowed to pray to their own gods.
Keep in mind that at present (1372/1373 DR) it is crime punishable by swift death (if you are lucky) to be a member of Black network and worship any god but Bane.

And finally - dugadar and drow. Are these races ever allies?

Hard to say. That depend on current situation.