im thinking headphone jack on mixer to microphone (that wire being 1/2" to 1/4" and stereo) input on computer, screw a little bit with the levels to get the volume right though.

maybe those tape outputs on the mixer? i dont know, depends on your available wires. but im pretty damn sure it goes in the mic input on your computer
you should have gone with an interface....mixers are more for live sound but anyways

get yourself a dual mono to stereo cord:

and two mono 1/8 - 1/4" adapters http://www.radioshack.com/product/index.jsp?productId=2062456&cp=&parentPage=search

plug the two monos into your L and R main out on the mixer
Run a stereo 1/8" cord into your line in, (usually light blue in color) on the sound card and over to the jack on the Y cord at the mixer.
With you using a stock sound card you will have latency so when you play expect lots of delays which will become a challenge when doing more than one track recordings.
no you don't use the main outputs to a PC. Use the tape out, its stereo line level. Use An RCA to 1/8inch stereo cable to the computer line input. It will sound fine. If your PC is old, try not to run other stuff (including the internet) when you are using it. That will eliminate most latency issues.

Normally on your PC you want to turn monitor line in to OFF while recording. You can also then connect your PC line output to the tape in of mixer and play your computer back through your mixer. Ideally your mixer is then connected to a PA or studio monitors for live sound.

But its not a very good capable mixer.