Jim, you ask some really great questions, and this is a doozy. Theologians have pondered this and produced a wide variety of answers. Probably none of them truly adequate.
I hope to do a post on this question of does or can God answer prayer and also what are the limits of his sovereignty. As to that question of whether God's will is done or not, that to me is answered clearly in the Lord's Prayer, the model prayer that Jesus taught his followers to pray. It says: "Thy will be done on earth as it is in heaven." Why would Jesus instruct us to pray that if God's will was always done on earth. In fact, this describes rather precisely the distinction between the realm of God's direct rule (heaven) and where God does not have free reign (earth).
As to the question of answering prayer, I wrote a small book about this quite a few years ago called Quantum Prayer. Unfortunate title as it sounds like a bunch of the quantum whoey stuff about self improvement, etc. But the point I was trying to make in this is given quantum indeterminism, given things like quantum interactions, wave collapse, and all that scientists see the result of these coming from random action. When an atom decays is very random. But, if something significant is tied to it, like poor Schroedinger's cat, then it can have real world results.
When you combine the actions at this level with chaos theory which says minute changes in initial conditions can have an outsize impact on complex systems, you have the possibility of God's intervention in nature in a way that does not violate the laws of nature one bit.
The question is, is all that happens at that level random? Or, can it be influenced by a divine hand? How would we know? The only way we could tell if it was truly random or not is based on the results. Spin a roulette wheel a thousand times and it always comes up the same, you know something is up with the wheel. So it is in nature. When we see positive or helpful results, such as a person very ill becoming well, or an accident narrowly averted, or a freak rainstorm bringing desperately needed rain, it may be very well be due to randomness in the interaction of energies. Or, it could be some tweaking here and there in a subtle enough way to make a difference but not to make it clear that there is overt influence.
Of course, then there are the miracles where those interventions in nature are not subtle and can not be contained within the appearance of randomness, but that's a whole other subject.
