Is performing stunts in designated, safe areas that provide ambulance and safety services considered unlawful, even though performing stunts in itself is impermissible because it exposes one to peril?

Perhaps the questioner means "joyriding" when referring to "استعراض بالسيارات" (displaying with cars). If that is the case, or something similar, then the crucial point is the presence of risk to oneself or damage to property. If any of these are present, then "استعراض" (displaying) is not permissible, even if it is in designated areas and with medical emergency services available. This is because such measures do not completely eliminate the risks, nor do they justify undertaking them.