could writing down that you are Muslim hurt you any way?
No.
“Since its inception in 1890, UChicago has remained committed to educating extraordinary people regardless of race, gender, religion, or financial ability.”
You can find a similar disclaimer on all college/university websites, this particular example was from UChicago.
Not really. Most colleges do not see or care what religion you put down. I believe it is only collected for demographic purposes.