Ok, I need to point out something that most newcomers to this game don't realize. This universe is brand new. It's not like we're taking existing DC and Marvel histories and putting the universes together. The premise is that these universes have always been just one, and we're telling it from the beginning.
So, in essence, everything is still brand new. Batman and Superman only became active a couple of years ago in game time (Actually, we're probably not that far off from syncing game time with real time.) The X-Men are still a bunch of teenaged students and not really a team. There's no Justice League or Avengers, and Captain America was part of the JSA back in WWII.
In this case, I'll fill you in on what's happened in Gotham. Batman's been on the scene for only two years now, Robin for one. twy played Selina back in season 1, and while I think she was a cat burgler, being Catwoman hadn't really happened yet. Batman doesn't even know about her.
Bruce does know about Vicky Vale, but that's pretty much the extent of it. He knows she works at the Gotham Gazette because he owns it. They flirted once when he was giving her an interview, but that's about it. Bruce is currently dating Elektra.
As far as Frost and Ivy knowing each other, that's perfectly fine. No one has played either character before, so their origins are open. And I'd love to see how they knew each other and the gap between Pamela and Poison that led to her becoming Ivy. So you can have as much fun with those two as you want.
And you can apply for Emma. But since no one's ever expressed interest in playing her, and if you're not going to use her outside of interacting with Ivy, then you might as well just use her as an NPC. And if someone does come along wanting to play her, you might be able to get them on board with your arc and have someone to interact with. Now, if you're going to use Emma in her own adventures, then you're obviously right in wanting to apply for her and can disregard what I just said.