Toyota Repair: RPM fluctuates/engine dies/cat. converter gets super hot./sputters, toyota celica gt, vacuum leaks

Question
I have a 1982 Toyota Celica GT, 189762 miles, 22R, 4 cylinder, carburetor & manual...When i start it first time of the day the RPM fluctuates between 500-1000, sometimes it goes down all the way and die, sometimes it doesn't. It starts sputtering also while the RPM keeps fluctuating and the catalytic converter starts to get hot...After 20-25 minutes idling(normal temps already), the fluctuation & the sputtering are gone, but the catalytic converter is superhot red.(it burnt the carpet in my garage). I finally  can drive it without worrying it will die on me.....I changed the spark plugs, the wires, the distributor cap & the rotor, the ignition wire, the fuel & air filter...i also checked for vacuum leaks, crack or tear hoses and i didn't find any...i always clean/spray the carburetor & put some carbs additives everytime i fill up.. 3 backyard mechanics told me to overhaul/replace the carburetor or replace the AAP(auxiliary acceleration pump) diaphragm or replace the cat.converter & muffler..I had this Celica since 1982 & it's very reliable to me...Pls. Help...Thank you so very much & i love reading your answers...You're a big help to so many.

Answer
The AAP may be leaking causing the air/fuel mixture to go rich so yes it does sound like a carburetor problem, also check the base of the carburetor for leaks, make sure there are no exhaust leaks especially at the bottom of the exhaust manifold where the pipe attaches, the pipes used to crack a lot. I wouold not replace the cat at this time unless absolutely sure the problem is caused by something else, the cat is just reacting to what's going on with the engine.