Other mistake: In the final scene both at the start when Fletcher is walking on stage and in a shot during "Caravan," just after Andrew mouths FU, when Fletcher puts a hand under his jacket the shot goes forward and then immediately shows the same footage in reverse. (01:27:55 - 01:32:45)
Other mistake: Fletcher first says to play bar 93, then after 20 seconds of playing and many bars later, he says to play bar 115, but the music sounds entirely different. If the bars were correct, the band would have already played the music rather than a completely new idea. (00:19:45)
Chosen answer: No, it is not normal to bleed while playing drums, as callouses develop on a drummer's hands through constant practising and playing. But, if a drummer plays hard enough and fast enough using the same drum sticks, friction between wood and skin can develop and rupture toughened skin. In the context of the film, the drummer is being pushed beyond his limits by the music conductor. Additionally, he is pushing himself to be better, so the bleeding is a consequence of that drive for excellence.
Scott215