If a supremely enlightened being can become many, walk through walls and dive into the ground as if it were water doesn't this mean that he is not dependent on his body and can actually make the body into anything he wants? If he can become many couldn't he just manifest a new body? I have yet to come across even one case of any of the supernormal powers that has been scientifically confirmed. Usually after doing some research on claims of these powers I find an explanation that completely debunks them as frauds. I don't want to know if these exist in order to gain them but because I want to be aware of any discrepancies with in Buddhist texts in order to better fully understand the Dharma. For me truth is the highest goal so separating fact from fiction is a must.
Then the real question is whether we humans are capable of seeing all the facts there are in the world? Carl Sagan put it best with his flatlander simile: http://www.youtube.com/watch?v=UnURElCzGc0 . Now say there's some being who possesses supernormal powers living in a cave on those extremely remote mountains, s/he'd probably has no interest to come down to our civilized world to prove anything. So in scientific term, it's a matter of the sample size. And in the words of another great scientist, Neil DeGrasse Tyson, don't dismiss the "whale" just yet: http://moniquill.tumblr.com/post/92432571651/claiming-there-is-no-other-life-in-the-universe-is
About the contradiction to old age, sickness,...well, there's no contradiction. According to the suttas and commentaries, the meditator has to meditate on a particular element kasina in order to be able to manipulate that element (ie. meditate on the water kasina to walk on water, on the earth kasina to go thru earth, etc.), in other words, s/he can only temporarily perform such feats under certain levels of concentration. They themselves are subjected to impermanence..
Even supernormal powers have its function. E.g. divine eye helps see things which cannot be seen by the normal eye. This does not mean that your eye will not be impermanent. Nor can any of these powers make anything impermanent permanent or exert absolute control over your corporeal body to repair decay due to old are. There may be some therapeutic benefits through but limited in effectiveness and application like any other treatment.
So supernormal powers cannot be used to overcome old age, sickness and death in absolute terms though there may be some therapeutic benefits in having high levels of Samadhi.
I'm not aware of any supernatural powers that can be attained through meditation.
An interesting scientific phenomenon comes to mind though. Birds are keenly aware of the Earth's magnetic field when migrating. They know which direction to go even without Google maps. Personally, I believe humans are capable of this as well but parts of our mind drown out these instincts. Dogs know when a tornado is approaching even without the Weather Channel. In a way, that seems unbelievable. It comes down to what you believe is supernatural. 1000 years ago a horseless carriage or penicillin may have been considered supernatural.
One time I did an experiment if I could tell where I was going by banging two keys together. I distinctly remember walking up to the wall and feeling the sound waves reverberate back without knowing the wall was there by sight or touch. Turns out this is called echo location and some blind children can navigate a playground by making a clicking sound with their mouth.
Initially, I would have thought echo location to be supernatural, but a scientific reason exists. Another example I can give is with aging. Most people would find it hard to believe that anything can reverse aging. However, meditating activates the parasympathetic nervous system. It is responsible for accelerating the process of digestion ("rest-and-digest") It's like Wolverine's natural healing factor in a way. If a person is in a steady state of rest for years, won't he or she be younger than those who haven't? You can notice what going two nights without rest feels like, and sleep deprivation is a cause of many fatal traffic accidents.
Another example, when I feel pain in my eyes from too much typing, I lie down and actually feel the pain. Initially, the pain sensation will intensify but then it subsides and then disappears altogether. I've heard Kusala Bhikshu say, "When you become one with the pain, there's no one to feel the pain." It certainly sounds supernatural, but there's a scientific explanation, and that's what Buddhism is, a scientific examination into suffering.