[awadecki]

That box in the air cleaner is definitely (as said before), the crankcase breather--the other half of the PCV system. You should probably replace it along with the PCV--it's only a few dollars.

Before you go about tearing through the carb, you should complete all of the ignition system repairs. Then--I'm surprised no one suggested this--you should change the fuel filter at the inlet on the carb. It's probably all gummed up. You might not even have a problem with the secondaries. My guess is that the carb probably isn't getting enough fuel.