I would need to go back to remember exactly, but were those Triune cultists or Lilith's cultists? Triune cultists originally worship the Three Prime Evils. Actually, the original form of the Triune cult worshipped the Primes without knowing they were evil.
I believe at some point Lilith co-opted the remaining Triune cultists into her order, they would probably still hold reverence to the original Primes in addition to her. I don't think the average cultist would have been aware of the rivalry between her and her father. If anything, they'd probably see her as being the Daughter of Hatred a kind of blessing.
I can only speak to the portions of the game I have played so far ( I just finished Main Mission 13, maybe 5-6 hours in total) but it's very linear so far. Each area is basically a straight path you follow, I honestly haven't even found many places where you can diverge from the path just for extra treasure or the like.
It could very well open up more later on like XIII did, but it's not open at all so far. There is a side quest system that could definitely lend itself to more of an open-world exploration feeling. I only have seen two sidequests that are given to you when that system is first introduced, so I can't speak as to how common they are yet.