If we can use the game as insight to the movie, I don't think he was talking to a cop. During the game (the level Boys in Blue I think), it's revealed that the Rogues killed a cop, and I think during an in-game cutscene it explains that's where Luther got the gun and badge. That soon after one of their own is killed, I don't think the cops would be likely to trust a Rogue.
With the calm, respectful, polite-ish way he was speaking, it might not have been a Lizzie. With the way we see him yelling and belittling his own Rogues and the Candy Store Girl, he probably wouldn't have been so respectful. I think the Lizzies heard about the shooting on the radio and took advantage of running into The Warriors. Alternatively, it is still possible it was a Lizzie- with the way he was speaking he might have been looking for attention and/or trying to impress her while not taking into consideration what 'Lizzie' is slang for.

Based on the "I dunno, I just like doing stuff like that" line, my guess would be that he has a mental illness that gives him a joy out of chaos, disorder and the pain and suffering of others. If Cyrus unites everyone, there's a big peace, which means less chaos and fighting going around. My guess would be that he's mostly likely talking to his girlfriend, his caseworker, or a Rogue higher-up. In all three cases he'd want to "check in" with the people and tell them what happened, however he was being vague in the conversation, meaning he might have been playing dumb/innocent to avoid telling them too much and getting in trouble. It's kinda hard to tell though if he was flirting (which would indicate a girlfriend) or if he was sucking up (indicating either of the other two).