Correct me if I am wrong (I am kinda learning all this myself), but DAC is a Digital to Analog Converter. Basically changes all the fancy pansy Binary 0's and 1's to an analog waveform. This waveform is nice and all but it is not "powerful" enough to pass the resistance, or impedance, of certain headphones which is why you need an amp. The amp helps drive that resistance of the drivers so that it can turn the waveform into ear-satisfying sound at an audible volume.
And just for clarity sake, you ALWAYS have an amplifier of some sort. The amp is the cricuit that provides the power needed to drive your transducers. It CAN be integrated, as it is in your phone, but a separate, dedicated amp has more room for better components. You always have a DAC, too (assuming a digital source, like a video game), and the same "dedicated components is better" rule applies.
You'll have to ask someone else about where the mixamp falls, though. I don't have any experience with them, and I don't need one since most of my gaming is done on PC. I assume it handles the "mixing" of surround sound down to two channels (two ears, two transducers) so you get 3D audio effects.