TOYOTA SEQUOIA STALLING

The first thing I would look for is a crankcase vacuum leak: at the valve cover, air intake hoses, oil intake cap, etc. The EGR valve could be clogged, or the EGR vacuum solenoid could be bad. The VSV (vacuum switch valve) could be bad. See:

Exhaust gasses recirculate into the valve lifter chamber / crank case to be reburned, to cut down on oxidated nitrogen and carbon that would otherwise be ejected from the tailpipe; when under acceleration, the high RPM creates a high vaccum in the system that must be regulated or the vacuum would suck too hard---- that regulation involves opening the system to intake a large volume of exhaust gasses. When slowing down and coming to a stop, the vacuum controls pinch off the exhaust gas intake. If there is a break in the vacuum system (hose came off, or is punctured), atmosphere gets sucked into the exhaust recirculation system that the controls cannot shut off: the vacuum cannot be made high enough to keep the engine's RPM high enough to idle properly.

Then again it could be something totally different. It could be as simple as adjusting the engine's idle.

My pickup stalls when coming to a stop and when the engine is hot: I suspected the positive crankcase ventalation valve (PCV valve) is stuck open, but now I think it is due to the huge hole in the air intake hose that I found three days ago.

I'd pop out the MAF and give it a good cleaning with some contact cleaner (available at Radio Shack, etc.) -- mis-read airflow can cause all kinds of issues. Cleaning the throttle body would also be a good idea, and easy to do.