all catalytic converter equipped cars have the air pump to feed air to the oxidyzing bed so yes, all 82s have it as it's part of the emissions control system. Do you have to elave it on? No not really but your cat won't function properly without it, only the reducing section will be operative and it shows in the emissions, I removed mine also, it's not a problem for the fuel injection system, the emissions stuff (AIR valves) is controlled, not monitored for proper operation.
If you want to haev a functioning cat con, yes then elave it on and the header tubes aid in O2 sensor warmup, you can do without them, just run the 1st switch valve to air cleaner and directly to the cat con, bypassing the 2nd valve (eliminating it) however it would be smart to install a heated sensor then, doing it in the collector would be a good choice (assuming you want to run headers, the only reason to do away w/ the air pipes)
The best way to activate the heater is to run it off the 12V power feed for the pump that comes off the oil pressure sender (when you start the car the relay momentarily powers the pump but shuts off, by then the oil pressure is up and the pressure sender/switch will have taken over the job of supplying 12V to the pump, it's a safety feature so the car won't run without oil pressure),that way the heater is only operative when the engine runs, not when the ign. is on (which it would be if you ran it off an ign. switched source...w/ a relay, always use a relay!!!)