What's up ya'll. Forgive me for being such a noob. I'm learning as I go so please bear with me.

I've been having issues with low idling and stalling when the weather is real hot or if I'm bogged down in stop and go traffic. My rig will fire right back up but I have to keep revving the engine to keep him going. To offset the issue, I had to adjust the fast idle air valve so my rig wouldn't become disabled. On a cool or cold day, my rig runs strong. I bought my rig in February and just recently got him running with a rebuilt engine so I'm trying to dial everything in.

I changed my air filter a couple of weeks ago and wondered why the HOSE DUCT (17882-89114) was designed to extend in and out of the lower air-box housing, why there is a male and female triangular notch on the lower air-box housing and the hose duct, and why the mouth of the hose duct is flared out? I noticed what looked like a Toyota part number on the hose duct and searched for it on-line and found this diagram. Apparently, my rig is missing the AIR CLEANER DUCT (53207-89107, not numbered but shown and verified by the local Toyota dealership) which apparently mounts behind the driver's side headlight and connects to the hose duct.

Correct me if I'm wrong please, but wouldn't sucking in the engine heat trapped in the engine bay be a bad thing? Wouldn't the air cleaner duct , which is missing, allow the intake of cooler outside air? Is this the possible cause for the idling and stalling issue or is it deeper than that?