Addressing your specific questions, it sounds like something is not set, or working, correctly. I run an iphone, ipad and macbook through the same set up you describe (I think - you're using a 3.5 mini jack to rca?) and all three play normally through several of my vintage and modern receivers when plugged into a line level (aux) jack. I run full volume out of the i-device and adjust volume with the amp or preamp. It sounds like any other line level input, i.e. tuner or cd player.
If you are needing to turn up the amp volume as high as you describe, something must be funky. The volume level imbedded in the streaming service cited above could be it, or have you tried using another line level input, like a tape input?
To the best of my knowledge a DAC is not intended to change the level of the signal; rather, it is designed to decode a digital signal. You are currently taking an analog signal from the iphone (converted from digital by the phone's internal DAC) and running it to the amplifier. You can also take a digital signal from the phone and convert it to analog externally, via the above mentioned docks and cd players for example. Most people here will tell you that the sound quality will be significantly better if you do so, as compared to taking the analog signal from the phone. So that may be something to explore, but it is a different issue from the question you asked.
My profuse apologies if I am stating a bunch of stuff that you already know, and that I misinterpreted your question in too simple a manner. My level of technical expertise is quite low, so it would be an easy error on my part. Further, almost everyone here can tell you much more about how to solve the specific problem. But it seemed like most people were responding with ways to improve the sound quality via digital out/ external dac, and the way I read your question you simply wanted to know why the analog out did not seem to be functioning correctly.
