Yes, that is a fox body mustang.
Will Turner is driving the Turner car in the second vid. Clearly, he has excellent car control skills. It was an exhibition lap, so that's why he is doing what he is doing I guess. He would go even faster if he drove straight. I wouldn't dare to try any of those stunts going in and out of corners at such speeds. It all looks so easy on the video with a pro driver. I bet the guy in the passenger seat wet his pants indeed.