Can I apply to student discount just with a school ID I'm an international student currently in an exchange program in a US school. I do have a student ID card, but that's it. 

According to your experience, will I be able to get the student discount? Does it take some kind of digital validation?

As I said before, I have a student ID card from my school, but I don't have a valid .edu address or anything. Does Apple perform some kind of checkup or just showing the ID is enough?
If your just looking for the ~5% student discount, yes your student ID card will be enough. 

For the Apple On Campus 12% discout, they will check from your student card if your school is registered to this program.