Your mare is very cute, and an appropriate hunter type. She's a pretty fair mover as well, and she seems to have a lovely attitude.
Please tell me how I can improve her movement and lower her head and collect her!
Her head and neck are high because she moves with a hollow/stiff back (pretty common in TBs). If you did some basic dressage flat work with her, got her working from her hind end more, and using her back, her head and neck carriage would improve and her already good movement would get even better. I would call that "connection" rather the collection; I like to reserve the word collection for its narrowest, dressage meaning, but lots of people do use it to mean moving through the back and traveling in a more united organized fashion like I suspect you want her to.
Beware! Trying to influence her head and neck with the bridle will likely make her stiffer and more hollow. Concentrate on the hind end and back, and the head and neck will come along.
You might want to consider getting a trainer to do some under saddle work with her to get her going in the right direction. Doing the kind of flat work that she needs while you're working on your equitation is going to be very difficult.
As for how she'd do in a hunter show, it's really difficult to say without knowing the level of competition at the particular show. Why not just take her for fun and see how you do?
Depending on the show and the judge, you could ask for some feedback from the judge or ask to see the judge's cards for comments. There's a way to do this properly - approach the show organizer first and ask. If the judge agrees to speak with you, be polite, listen
and don't get defensive. You can ask a question or two, but be aware of the time. Thank the judge and the show organizer.
BTW, I don't think you should have any regrets about buying her. She appears to be very suitable for you.