Yesterday I was reading through 'ask a theologian' which I have to say is probably the best thread on this forum. There was an interesting thing about the power of God, and one Plot's answers (apologies if I've got this wrong, Plot) was 'omnipotent means that if it is possible, God can do it'. So if curing an injury will inevitably mean you will die, God cannot cure it. Likewise he cannot create a rock that he cannot lift, because such a rock is impossible. Even more likewise, there's still a point in praying because he cannot tell the future, so he hasn't decided on the minutae of it.
On the question of 'why doesn't God heal amputees': If we say that God hears and answers prayers, then we have to think of how we see God. If we see him as the father, then it becomes quite clear. A priest once asked somebody, in response to 'if God loves us, why does he let us suffer', 'would you let your son go bungee-jumping if he wanted to?' 'Yes, father' 'even though he might be seriously hurt or even die, through no fault of his own' 'Yes father, but I don't see...'
The same logic holds. I don't think anybody here growing up, even if they had parents who had the means to do so (mine certainly didn't), was granted everything they asked for. You parents understood that living life knowing that you can always have what you want is no way to really live it. I don't pretend to know how God's mind works, but perhaps he wants them to be an example to others by their overcoming of obstacles?
Also, by curing them there would be no doubt that he existed, since only God could do that. One of his rules of engagement, if he's there, is that there will never be any proof of his existance. Otherwise, we wouldn't have total free will and would be hugely obliged to live a Christian life, knowing for sure that he was watching, and his great gift to us (apart from the whole redemption/Jesus business) is free will.