is that the brake pedal will vibrate when you brake, maybe accompanied by steering wheel vibration. You didn't mention pedal vibration, but describe shimmy when you are not braking. The pad heat thing, or the lip thing, doesn't make sense to me either. I think the problem is elsewhere, like steering rack bushings or other worn parts. What changes when it gets hot? Tires do. You could switch front and rear to see if anything changes.
