Yes and No
Running multiple mics into the mixer and then recording into your computer would require you to do one of two things. First of all, you are still connecting the mains on the mixer to your SB Live to record one stereo channel. The problem with this is that any effect that you want to apply to an individual track (one mic instead of all of them) will be virtually impossible. Since they are all part of one stereo channel, would have to retrack the entire song and tweak as desired. With the number of mistakes that I make while recording a track, I would never choose this method.
Another option that you have is to only record one mic/instrument at a time to seperate tracks on your computer. At least this way you have more flexibility in editing what you have recorded through software and plug-ins. This requires that after recording each track, you must connect the next instrument to be recorded and start all over again.
Getting a better soundcard that has multiple inputs/outputs would cure this problem in a heartbeat since you can now connect each instrument to it's own channel on the mixer to a a seperate track in your software in one recording. Hope this helps a little.
Dick