I know this is going to seem like a repeated post but here it goes.
I've had my CA in my S13 for about 4-5 years now, and I've always had problems with the cold start. It wasn't until recently that I started to dig into the car looking for a cause for my high idle, my ignition timing being all over the place, and having no power, and NO COLD START. But I have never have had cold start, even after replacing the air regulator and IACV.
As I was browsing the internet for a solution to my random high idle and my ignition timing issues, I came across a post on an Austrailian forum saying something about the 10A fuse (engine cont.) in the passenger side fuse/relay box. MY WAS BLOWN!
After replacing the fuse, BAM! I now have cold start. I never thought to even check a fuse for this issue but it sure did fix a lot of issues. Without that fuse, the AAC valve on the IACV does not get the proper voltage to power the valve, as well as quite a few other things in the whole ECCS system.
My lesson, check that fuse before testing and tracking down parts and giving yourself a headache..It may not be the case for everyone, but I wanted to share my experience.
