At one point towards the end, Ziegler says to Bill...

"that if he knew those names, he'd lose sleep or something like this, because he wouldn't feel safe."

Because they are people in positions of power, who, like Ziegler, can influence others, manipulate others (as Ziegler is doing with Bill), have people bashed up, raped, murdered, disappeared, can mysteriously have access to information that nobody else does ...

"When I paused at the newspaper article about the murder of Amanda/Domino"

The woman referred to, Amanda Curran, was the "Mandy" who OD'd at Ziegler's party two nights previously and who may also be the "Mysterious Woman" who Bill meets at the Somerton rituals, Bill's 'redeemer'. Domino was someone else entirely, a college student reduced to prostitution to "pay the bills" ("Bill": the name of the protagonist, dollar bills, bills to be paid, bills to be collected - Ziegler to Bill at Ziegler's party: "I only had to look at his bill"; Sally to Bill at Domino's apartment: "You're Bill!? THE Bill!?") who Bill meets on the street shortly after he leaves Marion's apartment, the daughter of the Senator who has just died, and just after he's assaulted and taunted by the gang of aggressive, macho-posturing, sexist college kids.

"I suspected the police might be into it too, because their behaviour seemed weird. They were adamant that this wasn't a murder even though she was seen with two other men at 4 am, much like in Nick's case."

But this is Ziegler's (unreliable) account, what Ziegler tells to Bill when they meet in the poolroom. It totally contradicts what is in the newspaper report that Bill reads. How would Ziegler know that "the police are happy", that "the door was locked from the inside", and so on? Is he just making all of this up to placate and silence Bill, to shut him up with a combination of implied threats and intimidation paradoxically mixed with a trivialization of all that has happened ("It's all just a charade", "Nothing really happened", "Life goes on"), or does he have privileged access to internal police data and investigation, knows the police commissioner or some other senior officer, because the latter was one of the guests at Somerton?

The participants are definitely the upper-upper class like Politicians, Bankers, and Industrialists (?) Bill even though rich is not in the same league as them and Ziegler.

I think Ziegler definitely had Nick and Mandy murdered to keep quiet and knew what was going on with both Police and Media. That was probably the scariest moment in the film actually and Bill realized he was pretty damn close to death.