She sounds very bright and keen. Are you sure that when you're explaining stuff to her, you're not pitching it at a level that's way below her ability and/or understanding? There's nothing worse for a bright worker than having stuff explained to them at a level that a year 10 could understand and this would explain her "jumping in" before you've finished. She could just be a couple of steps ahead of you.
You've already said that her work is great, it may be that she's doing stuff that's way below her level of competence.And the "involving herself" in stuff that's outside her remit smacks to me of someone who's bored and under-occupied and thinks she's showing initiative.
It also sounds like she hasn't got enough to do if she's got time to snoop on Linked In etc about potential hires that are nothing to do with her.
Keep her busy, and give her stuff to do that will challenge her a bit.
You could always try sending her off on some errand or other when she tries to get involved in your discussions with more senior managers. Or just say "Did you want me, X, only I need a private chat with Y just now" and arrange to catch up with her in a minute.
I also wonder if there's a bit of sexism here, OP. Go back and re-read your first post, but use the male pronoun. It sounds as though a lot of her behaviours would be regarded as positives, not negatives, if she was male, and women are often criticised for the sort of "pushiness" that men are rewarded for. Things like "confident" and "forward" aren't things to be down on a staff member about imo. Would you really want them to be timid and fearful of contributing?
Do you have regular 1 to 1s as part of your induction for new staff? If not, you need to start. They are an excellent opportunity to highlight to staff the need to fit in with the organisational culture, and to address things like sticking to her own work and not trying to get involved in other people's. She might have come from a working environment that's much more informal and collaborative, and where junior staff are encouraged to have input into everything.
You definitely need to address the timekeeping, but be sure that everyone else is on time too, or it'll look like she's being singled out.
And speak to your manager about how to address this - they're there to support you, too.