First, full disclosure: I can't imagine being with someone and not wanting to know about their past relationships. A person's past, including their romantic past, made them the person they are. Getting to know someone, for me, means knowing about the people and relationships that shaped them. I also understand that there are people who do not feel this way, and prefer to know as little as possible about their partners' pasts. Nonetheless, I wanted to give full disclosure about where I'm coming from because I think it colors my advice.
Your problem isn't really about whether you want to know the details of her past: you want to know if you're a rebound. If you continue to be a part of Charlotte's life, you're almost certainly going to learn stuff about her last relationship - I mean, she still has a house with this person; it's going to be very hard to avoid knowing anything about him, since they're still in the process of disentangling their lives - but that's a very different question than whether she's over him enough to be with you. For that, only time will tell. If you're having fun for now, perhaps wait and see how things develop. If you're worried about getting hurt and it's interfering with your ability to enjoy the new relationship, I think it's totally reasonable to walk away (for what it's worth, I think it's likely that she's not yet over her ex, but hey, I could be very wrong).