Spoilers time!
After you witness Tchortists stealing cube you know it's no longer in Core City.
"Hi guys, you can move on since item you're looking for is no longer... what the heck?! You wanna fight? Great choice as i am the Invictus and Invictus never runs away!"
My interpretation of conversation with Faceless in West Wing and elevator fight:
Faceless: "Hi player. We're going to kill everyone in this place including you if you won't leave."
Player: "Why, thanks for the warning, man. But i'd rather stay here to fight hand in hand with your archenemy and kill as many of you as i can before proceeding with my business in Deep Caverns."
The Faceless shooting on sight is a different issue, though. AFAIK, lore-wise the Faceless are supposed to be somewhat unpredictable: I think it's Old Jonas who advises the player to steer clear of small bands of Faceless, since those tend to be more dangerous/prone to attacking than larger groups. That said, if you go through the Oligarchy questline and only after that speak with the Mindreader in Rail Crossing... That should probably have a pretty big impact on the Faceless, and I'm not sure if the game recognizes this possibility at all.
As for the friendly Faceless... well, yeah. You won't encounter him if you didn't help him in Core City, but if you did do that, it does make the whole Institute assault and elevator fight pretty weird. Giving the player a choice to temporarily side with the Faceless, or leave and either hide during the battle or fight alongside the Tchortists (hostile Faceless, but if you don't attack them you don't get any penalty to your reputation) would probably make more sense.
This choice could even be the fourth main deciding factor in whether or not the Faceless in DC decide to ally with the player. If you screwed up in Rail Crossing, but helped the Faceless near Foundry and in Core City and then fought the Tchortists, the Mindreader would still consider you as a friend. You could even go back to the Tchortists in DC, since as far as I remember, nobody actually made it down from the Institute apart from the player, and thus the Cytosine Outpost garrison wouldn't have any idea you betrayed them before.