A Station Eight Fan Web Site
One aspect that I always liked about "Gargoyles" was that most of the individual characters had someone they considered their own personal nemesis.
Brooklyn had Demona (but I doubt she thought enough about him to feel the same, her focus was always on Goliath). Lexington had the Pack (though, again, I doubt they gave him as much thought as again, Goliath). Broadway had Dracon (again, I tend to think Dracon gave Goliath more thought, and especially Elisa).
Actually, while writing this post, it just occurred to me that these people the trio despised probably didn't give them individually all that much thought... kind of says something about how the futility of holding such contempt for someone, only for that other person to probably not spend all that much time thinking about you. See, I love "Gargoyles," I'm always seeing things in new ways.
But I guess what I was originally going to ask, before my little revelation there was this. Did you have any rivalries like that planned with the Redemption Squad?
For Hunter, Demona or John Castaway being her personal nemesis seems like a given. Though I wouldn't be surprised if we get a new character there. Dingo seems to have Falstaff for that role, especially if he ever discovers how his mother died.
Yama, Fang and especially Matrix are harder to pin down for this. I suppose for Yama it could be Taro, but I'm not sure... he seems to blame himself for what he did rather than Taro. Fang doesn't seem to hold any real ill will towards Sevarius. And Matrix doesn't even have emotions and can't take things personally.
Of course, the ultimate example of what you describe in your third paragraph is with Gillecomgain. He's obsessed for years over the gargoyle that scarred him. And when Demona sees his face, she has absolutely no memory of the incident.
As for the Bad Guys characters, things would evolve in the fullness of time.