Nice post, OP.
I do believe God answers prayers. Not the way we may want or expect, but he answers prayers.
My biggest prayer story happened about five years ago. DH had had a vasectomy 2 years previous in 2000, and I had regretted it and prayed that God would allow me to have another baby (with my DH

). I found out that I was, indeed, expecting in April of 2002. We were both overjoyed.
In July we discovered that we were having another little girl. She was "waving" at us in the ultrasound.

Then on August 14th, at a routine 20 week checkup, we discovered that our little prayed for baby had died. I had labor induced and she was delivered on August 16, 2002.
I cannot begin even to describe the anguish of losing her. It literally was, and still is, a physical pain. But, my DH and I also had to deal with the very real mental/emotional pain of feeling,at first, that God had betrayed us. After some time and soul searching, we accepted that even though we lost her, her short time with us, both before and after she was born, had touched our lives. Now, I'm not saying that we were blessed to have lost her, but we were blessed in different ways for having had her, even as briefly as we did.
So God did answer my prayer in that He allowed me to have another baby, but He didn't allow me to keep her. Now, I know critics will say that sounds like a cruel joke on the part of God. I don't see it that way. I see it as God answering my prayer as He saw He needed to rather than the way I wanted him to.
Naturally, I would have preferred that God let me keep her here for awhile. But, that was not in His plans. I believe that He did that for both reasons that I believe and ones that I don't yet even know.
Seven months after I lost Gabrielle, I became pregnant with Olivia, and despite a few scares, she was born healthy on Nov 25, 2003. So ultimately, in His own time, He did answer my prayer the way I had hoped.
Sorry so long and I obviously know that it won't be considered a miracle, an answered prayer, or anything related to God by some.
If you read this far, I congratulate you on your persistence!!