1.) Because Rohan is the protagonist and we are meant to somehow sympathize with him more through this scene. Also, I think the principal was probably aware of how harsh Rohan's father was to him.
2.) My guess is that Arjun is too young and physically incapable of keeping up with his father, whereas Rohan could do it if he really pushed himself.
3.) It was never insinuated that Arjun had a medical condition, although he was certainly frail. What happened was, as Arjun later mentioned, he broke another kid's teeth in a fight that the other kid started. Arjun was probably picked on before and retaliated in the same way, so the recurrence became an issue (even though this defensive violence was most likely a learned behavior from his father).
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Because Rohan is the protagonist and we are meant to somehow sympathize with him more through this scene. Also, I think the principal was probably aware of how harsh Rohan's father was to him.
I'm sure the principal saying sorry was not a means to get the audience to sympathize with Rohan. We learn later in the film that Rohan's father does come to visit the boarding school and meets with the principal, so the principal surely knows the kind of person he is. You can sense from his expressions that he really did not want to expel Rohan, but he has to, because this was the third time the boys did something bad, and as a principal he has to follow the school's rules. It can be assumed he also knows Rohan was a very capable student, and he probably knows of his writing talents too. He also must've known that the father did not actually meet Rohan for 8 years.