I'm probably going to bring down the wrath of ehell here, but here goes ...
First, let me stress that DD absolutely doesn't have to go to the prom with Jay or pay for the ticket or anything else. I don't think she did anything wrong, although she could have been clearer. If I were her mom, I'd advise her to tell Jay what others have advised, something like, "I'm really sorry about the misunderstanding, but I won't be going to the prom. I hope you have a good time." Not because she has anything to apologize for it, just because it's the kind way to handle it.
And I understand that here at ehell, we all want to be supportive of the OPs and take their recounting of events as absolutely accurate and all ambiguities to be resolved definitively in the OPs' favor.
My concern is the rush that so many posters have to label this boy as a creeper, a stalker, a manipulator, potentially dangerous, going to lie about her to others, going to blame women for everything forever, and I don't know what all else, and to assume all kinds of facts that, rereading the OP's posts, aren't even there. For example: several people take Jay to task for "getting his mother to call the OP" -- maybe I missed it, but I didn't see anything at all that suggested that he even knew she was calling the OP. It is at least as likely that he would be totally mortified if he knew.
I don't say -- at all!!!-- that either the OP or her daughter is lying or misrepresenting the exchanges. But I had teenagers at home not all that long ago -- a boy and a girl -- and I know that they don't always remember and recount things exactly as they happened. Adults, too. It's human nature that we remember things in the best light for ourselves (have you ever heard two people recount the same argument? It's a staple of sitcoms). The conversations in this story may not have been precisely as we are hearing them third-hand, and we don't know any other context, either.
Another assumption some posters made is that when she said "okay" in the second conversation, it was clearly an "...okay?" that meant "huh?", not "Oh, well, okay then" -- or even that it reasonably could have sounded like that to an optimistic teenage boy. At best, it certainly wasn't an adequate way to communicate, "No, I'm sorry for any misunderstanding, but as I told you, I will not be going to the prom" -- which she could have said in just those words.
I think that his hearing her "okay" as the "yes" he was hoping for is quite possible because she gave him the reason that she was babysitting the first time -- not that that wasn't okay (I think it was great), but it gives a clue to how he would interpret "okay." As we have often discussed here, sometimes that sounds to people as if you mean, "I would love to, but there's this impediment, but if I can get out of it, I might." (This is a very typical problem that people have with friends who have trouble saying no -- they think that they should help solve the stated problem.) That's why we tell people to be polite but clear when declining, as several people have suggested as the takeaway lesson for DD, too.
Like others, I feel that of course DD has every right to have any rules she wants for her dates. But also like others, I think that it is really unrealistic for her to expect others to know that she doesn't consider something a date unless the guy asks her father for permission. I have never seen that even once in my own or my kids' teen years -- frankly, to me, that would seem a little off -- it wouldn't feel so much like courtliness as make me wonder what level of importance this kid was expecting this date to have. If he asked her dad before even asking her, that would seem very presumptuous and even kind of creepy to me, but there's nothing that suggests that that's what she means. Well, I hope not -- but she said that she didn't think it was a real request for a date because he hadn't asked her father, right? So wouldn't that have to have happened first? Or possibly she only said that later to her mom when she was justifying herself?
Let me repeat, if that's the way she wants to approach dating, fine. I'm only bringing it up as an example of something of a disconnect between her and Jay at the get-go -- a pretty major one, in fact.
So why assume that Jay is a horrible sicko, when it's just as possible that these are both inexperienced kids who just didn't listen to each other well and didn't communicate clearly, maybe not equally, but so what? Does someone have to be the bad guy in every situation?
As someone else pointed out upthread, we often say that it's important not to assume the worst motives. Especially, I would think, with children.