Why would you check the alternator?
Check your head temp sensor. If that is bad or doing something weird that could also affect the cold idle. The ECU reads data from that, the thermotime switch, and the air temp sensor [located in the AFM] to calculate fuel. Read the EFI Bible. You can find it on a quick search here, it's actually pretty useful. I would recommend spraying the all the connectors with Electric connector cleaner.
If the Air Regulator isn't functioning properly, here's what I did for my car:
The car isn't getting enough air. If you set the idle higher, you won't need to hold the throttle down. The down side of that is that it idles really high after it's warmed up. I have an unused vacuum hose running through my firewall into the cabin. When the car is cold, I keep it open, which tricks the AFM by giving the motor unmetered air. When the car warms up, I plug it off and the idle drops back down.