You've got it. DACs convert to analog, and they can come in packages with amps, but they might not (and many people prefer keeping their analog bits separated from their digital ones for noise purposes).
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.