The other question I have is does setting up specific party members on the quests do anything like increase the chance the men of that group will likely have scenes with the girls or something?
Yes and No. The game doesn't really tell you but to get specific NTR scenes, each Heroine has a set character they have a "route" that they cheat
Kuno (red hair girl) [MC's fiancé] - Mercenary couple with pink hair priestess (can't remember his name lol)
Yellow hair elf - The playboy mercenary
blue hair Idol "siren" - Old dude mercenary
You essentially set them up at the mission board next to blond girl in the center
These missions have set requirements for a "scene" to happen. Which is stated in the board you can check (no idea how bad the MTL botches this tho)
These mission boosts the stats for the girl X cucker.
You can set them up with different sets like Kuno X old dude, but you just get text without CG NTR.
--edit---
Played it again because It's been awhile since I last played the game, just some minor fix.
-You set them up with the blond receptionist 「ルイ」(Rui) not at the board.
-The board is for the Brothel post for the girls (except Uka who works at the bar if you post her there)
-The mission condition for NTR scene guide is at the far right bookshelf at the common area (Where Kuno and Uka studies)