I am going to say that the answer is most likely no. I googled this extensively! This is what I had found regarding them regulating their body temperature: As cats are familiar and easily kept animals, their physiology has been particularly well studied; it generally resembles that of other carnivorous mammals but displays several unusual features probably attributable to cats' descent from desert-dwelling species. For instance, cats are able to tolerate quite high temperatures: humans generally start to feel uncomfortable when their skin temperature passes about 44.5 °C (112 °F), but cats show no discomfort until their skin reaches around 52 °C (126 °F), and can tolerate temperatures of up to 56 °C (133 °F) if they have access to water. Cats conserve heat by reducing the flow of blood to their skin and lose heat by evaporation through their mouth. They do not sweat and pant only at very high temperatures. Unusually, a cat's body temperature does not vary throughout the day; this is part of cats' general lack of circadian rhythms and may reflect their tendency to be active at both day and night. Cats' feces are usually dry and their urine is also highly concentrated, both of which are adaptations that allow cats to retain as much fluid as possible. Their kidneys are so efficient that cats can survive on a diet consisting only of meat, with no additional water, and can even rehydrate by drinking seawater.
That was taken directly from wikipedia.
As far as why cats purr, they do it to show extremes of emotion. They will purr when they are scared, angry, extremely content and happy. There is some belief that the reason cats purr when scared or angry is their way to comfort themselves, kind of like us singing to ourselves to comfort ourselves. I'm not so sure, but I thought it was interesting nonetheless. They may also purr when they are sick.