R., This is such a common question! I don't have any easy answers, but here are some thoughts that I hope will help. It is possible that your daughter has had a painful BM in the past that is causing the problem... or perhaps she has some pain or discomfort with each one that is scaring her. There is also the theory that for small children, the very nature of the poop makes it seem to the child that they are losing part of their body when they go, and for some it's the flushing that away that is frightening.
You mention her having a problem with the noise of the toilet flushing. I can recall when I was about that age that any running water under me was a frightening experience. I'm old enough to remember not having a flush toilet (used the old fashioned outhouse) but recall an incident where I was sitting on the cover of the cistern on our farm talking with my father. The water from the well began flowing into the cistern beneath us, and I freaked out. It felt to me like I might fall into the cistern. It was a completely foundless fear, as the cover on the cistern was secure, and though I knew that in my mind, my emotions told me I was in danger. My father held me close and kept telling me I was OK, and couldn't fall in. He made me stay there until the water stopped flowing in. Though I recall being terrified the whole time, I believe his method of making me sit there and find out that I was OK helped me get over that fear. You might find a similar way to help your daughter get over her fear of the toilet. I know we want to protect our kids from frightening things, but I also think that sometimes we just need to help them through them rather than keep them from them.
There are no easy answers, and you'll need to keep on reassuring your daughter, and encourage her until she gets it. And though it may seem hopeless at this time, I assure you it won't be long until she has it mastered.