What is the ruling on delaying the Fajr prayer until after sunrise due to overwhelming and recurring need to relieve oneself, and the emission of a small amount of blood from scratching due to pubic hair growth? Is there a sin incurred for this?

You erred by delaying the prayer until its time expired, even though there was ample time to perform it. Do not repeat this action, and be diligent in performing prayers on time. The reason for the delay is the excessive whispers (waswas) that you must struggle against and strive to eliminate. As for a small amount of blood, it is excused and does not invalidate ablution.