Sounds like the engine maybe flooding or rich under the warm start condition.

You say you have to give it air by stepping on the pedal. The engine should start without doing anything but turning the key. The idle control motor sets the idle speed via the computers mapping program. The motors are frequently the problem. They get sticky. They rarely can be cleaned. If it has 80,000 or better its worn. That causes late idle reaction but will also or can cause problems when cold....now it gets tougher. Rich conditions can be cause by leaking cold start injector or just one injector. If it sets longer the extra fuel could just vaporize. If engine is cold extra fuel doesn't hurt. Injectors are fairly reliable. If it starts cold and doesn't stumble to catch up then the rest pressure on the injectors is not falling off. That means they are probably OK and so is the check valve on the main pump.

I like the sticking idle control motor best because I've changed them on several cars. The coil is out there,too. Ignition is the place to start and you have covered that darn good. If coils get too hot that could give a weak spark and bad firing. It could be just a hot thing but why 10 to 15 min. Heat soak? May, if you know it is going to do it, put a wet rag around it and cool it. If it stops it, right then, you got it.

Another possibillty is the air mass meter is getting to hot due to a bad thermostat in the air cleaner. Not letting enough outside air in than exhaust anifold heated air. That can whack meters out of calibration. Its not cheap those meters. Check and make sure the doors opens (at least partially) when the engine is normal temp. I lost a meter because of that 10.00 item! My car is an 86 though, I'm not positive an 84 has them. Thats all I can think of GOOD LUCK!