No, that mixer does not have a USB nor FireWire computer interface. I was just getting something similar that has a built-in USB or FireWire connectivity. There are now numerous similar units to that 1202 and have that capability. But that capability, you don't need to purchase any kind of external computer audio interface. But beware, most of them that have a USB connection generally don't handle more than 2 tracks of record at a time. That's not to say you can't mix for microphones together to a single track or a pair. The devices feature FireWire connectivity frequently have the ability to record 8 or more channels simultaneously to individual tracks. This is much more advantageous when you are trying to record drums to multiple tracks simultaneously. Sure, you can create a two track mix but that's not quite the route you want to go. So even something like one of the PreSonus units or Mackie Onyx, plenty of others similar to those. Now those don't necessarily have any kind of actual mixer features. They're basically just an input, output device and the mixer is in the computer. Whereas the Mackie 1640 is both a standalone analog mixer & multitrack computer FireWire interface. And there are other similar units to that by other manufacturers, all decent.
It appears that you are good to go on the microphone end of things. Not my favorite selection you have there but all doable. So to record your drums, you may want to record 8 or more tracks simultaneously and that's why the Mackie 1640 might be more up your alley? Great for live work, studio work, all that. They also have their Onyx devices which supposedly are sonically superior but hey, I never met a Mackie I couldn't use.
Time to beat off and a 1, two, three and...
Mx. Remy Ann David