If he is concerned about the image of his family, it is possible he wouldn't want to jeopardize his family. Even if he's not actually cheating, the staring, and having her staring back is still disrespectful and he should know that. It's hard to know how to act when someone shows attraction towards you, even if its out of line, there's still that bit of fascination about the unknown, but he realize that if you noticed it, then others may notice as well. Kids can be cruel and if this were to get around the school there's no telling how they might treat your daughter.
The best you can do is trust him until he shows he can't be trusted....you can't accuse him of something he hasn't done, but you can let him know that you're aware of the looks whether he wants to admit to them or not and that he has to understand that by looking at her he's giving her ideas or at the very least the idea that he might be interested even if he's not.
Getting to know her better. I'm not sure what you mean by that, but I would never bring her to your home or near your family. It's true that you should keep your friends close but your enemies closer, but be careful of making friends with her just for this reason. Being friends with you could give her the perfect excuse to contact your husband.
I welcome your thoughts, let me know if you want to talk more