The child is looking at either a later model Hyundai Elantra or a Chevy Cruze. Something <$10K, since I'm helping pay for it. I keep weighing the idea of buying someone else's problems and suddenly making them mine. I'm not in a position to buy her something new, and what you can buy CPO or new for <$15K isn't exactly appealing.
You've gotten some good advice by other members on this thread, but one of the first things you stated is the most obvious thing that people forget....that is, that a vehicle that you're going to find for under 10 grand will be vehicle you have no real record of, might have at least two previous owners...and will most likely be a higher mileage vehicle, also!
So, let's look at it like this: at least you know, say if you put in another $1500. or so( I'd personally just forget about the exterior damage at this point...trust me, she' ll never take care of it the way you did, anyway!
) that it is a fairly solid, one owner, well maintained vehicle that should last her a few years, if she does the maintenance.
Do you think you'll likely find that in a used car under $10k?? I would say...no.
Keep in mind: the things you mention are not big dollars items, really...I could see if it needed a new transmission or all new struts or something of that nature.
But really, issues like doors switches, and maintenance items like brake fluid changes, serpentine belts...are relatively easy fixes by a qualified tech.
And, as someone else said, it's a "college beater" anyway.