I ran mine without the secondary air system for 5 years as it was broken (the fan was melted and rusted to death, I removed it completely and capped the hose inlet to the pipes) without a problem, its only there to speed up the cat warming up so won't effect running, just emissions when cold. The intake air temp might cause problems, though the popping sounds more like a misfire, I'd suspect its more likely the o2 sensor in the exhaust if it was a sensor (given the popping there is the possibility thats been damaged anyway, probably what also took out your cat).
The intake air temp sensor is either in the MAF, or in the tube that leads between it and the manifold, can't remember which as no longer have that car. If its in the maf then unplugging it should cause it to use a default map (Though an error condition should cause the same thing to happen anyway).
Can you see any live data? If your using a bluetooth adaptor then Torque with an android phone will display this, I'd have a look at the o2 sesnsor reading (1x1) and see if your getting a wave output shown. If its dead (same reading all the time) then it may be that causing the problem, equally the MAF might be causing the problem, though that usually causes lean running. Coil packs tend to be more temperature sensitive and likely to throw intermittent faults when starting to fail, equally the ht leads can play up in the same fashion.