Does "she" refer to "mouth" here? Is it common to say "she" as a body part?
Yes, he is referring to his "mouth" here. It's not common to refer to body parts with pronouns, but the speaker has done so here to be humorous and also to express how important his mouth is to him. Usually, body parts are referred to by name or "it." Ex: My arm hurts. Don't touch it. Ex: My face feels hot. Does it look red?