Man, I have been hanging around Corvette Forum too much lately. The first thoughts in my mind were "pics of daughter?" lol
It is a Corolla. It is possibly one of the most reliable if not the most reliable cars built, especially with only 4000 miles on it.
Fluids, belts if old, the timing belt isn't due until 50,000 or some years. It might be worth rolling it through your local Toyota dealer if you are in good standing to see what they have to say about the timing belt, or just stop by and ask.
Check the tires. With 4000 miles, I assume they are original, but at 4 years old and so few miles, they could be dry rot from the heat. If she is doing a lot of highway driving in Texas heat between school and home, you might want to replace the tires before she goes even if they look good, because old ones with low miles can come apart when finally strained in Texas heat.
Battery is probably dead or dying. If it still has the original, replace it and be sure. Toyota was putting Delco batteries in their cars and I found they die almost to the day of the car being 4 years old, which I believe was the coverage on the battery, too. In most cases, you could let it go, but a young girl alone at college will have it die at the most inopportune time possible. So if it is close, preempt it.
Change to the insurance. Cars on campus can be targets and while you are probably saving money at home with only liability, the risk goes up significantly there and you could lose it all.
I agree with changing the title and insurance to her name if she is 18. When I got my car, it was all in my name and it was one of the greatest favors I ever did my parents. I don't think they really knew how much. Although years later, my brother got them cancelled and out of the company they had been with for 25 years, although it didn't affect me.
The no-car-at-college-the-first-year is a good one. The kid doesn't think so, but it does allow them to concentrate on school and not out driving around. Of course, they might need it for a job or to travel back and forth, but if not, it is better with none.
Good luck to her.