i was just looking over this forum for an answer to a differnet pb display problem.
i'm not sure which hinge or which corner you are talking about, left right etc., but in any case the hinges can be loosened so that they give you a smoother action. just get ahold of a T7 torx and a small philips head screewdriver.
remove the hinge cover with the torx 7 and then carefully turn one or two of the black (they are on mine) philips head screws counterclockwise to loosen the hinge just enough to let you open and close the cover properly. now replace the hinge cover.
as far as the machine not booting up, i doubt that the hinge would influence this unless both problems are related to a serious shock to the pb itself, i.e. falling of the desk. but you would likely see signs of that.