How does Mustang fight in the end of Fullmetal Alchemist with injured hands? Somewhat trivial, but here goes: in Fullmetal Alchemist, we see Mustang's hands get pinned down by Bradley's swords when Wrath and Pride are trying to force him to do human transmutation and open the gate. Given this, how does he manage to participate in the fight against Father later (probably in chapter 107 or a slightly earlier chapter) given that his "attack" requires use of his hands (and that he doesn't seem to start using circleless transmutation until he has to fend off a counterattack)?
This is just a scenario of the shounen genre doing whatever it feels like. Biologically, he would have taken serious damage to bones, nerves, muscles, etc. when his hands were stabbed through. But in anime, particularly in the shounen genre, characters can often overcome this type of thing with relative ease.

Take, for example, a scene from Shingeki no Kyojin (Attack on Titan) wherein one of the characters literally has a plank stabbed through a lung, and still manages to somehow sit up—dragging the plank through their body—while spurting blood everywhere.

In Mustang's case, there is no evidence that he was healed in any fashion (such as by May Chang), nor that he is impervious to hand pain in any way. He is simply able to do this because the mangaka decided it was necessary to the story. That's the power of shounen!