Can I link a pc to a powered mixer system?
There are specific programs and hardware to make this really work well or you can hard wire one in as well. Use a mini to RCA adapter to a input to the mixer. You really need to talk to a pro though for anything better.
Best mixer around 8-12 bands for a rock/punk band?
Powered or unpowered? Unpowered, you can't beat the price on the Behringer Xenyx series or the Peavey PV14. If you're going powered the Carvin 880 is great. The new Behringer PMP3000 or 5000 are supposed to be a big improvement on their earlier mixer...
Condensing guitar, bass, vocals, and drums through mixer into Eon amps: good idea or bad idea?
You mean EON powered speakers? If so, then sure -that's what the mixer is for. Just make sure it's NOT a powered mixer. You connect the LINE output to the speakers. If its a stereo mixer, you hook up left and right and then you can pan.