You're wasting your time running a new cable from the alternator to the battery, as long as the battery is remaining charged (you'd know in a day) then it is working. Besides, high voltage will cause that problem, not low voltage.
There are a lot of things to check here that might not have been checked at the dealer or other shops. The throttle plate could have been adjusted wrong, or might be sticking. The throttle plate stop screw, which some people think is an idle screw, is not supposed to be adjusted. Perhaps in Romania you aren't used to stupid people who think they know everything, and do things to cars that cause problems like this because they don't know everything. I've seen plenty of stupid Americans who do, and they use screwdrivers on the throttle plate stop screws. Remove the air intake so you can see the throttle plate. Work the linkage by hand to see if it is sticking and not shutting all the way. Look closely at the springs and cable attachments to see if anything looks out of place. If you can't tell if the throttle plate is stopping where it should, then try loosening on the throttle plate stop screw, and see if the throttle plate moves backwards. There is a lip inside the throttle body that stops the throttle plate, that screw should stop the plate at the same instant as it hits that lip. To reset this you'll have to have a good sense of feeling- loosen the screw until the plate is against the seat. Then use one finger on the plate while the other hand slowly turns the stop screw back down. Set it so that the stop screw stops the throttle plate at the same instant that it touches the lip on the inside.
Of course, if you find that the cable is sticky, and holding the throttle plate open- replace the cable. You can try lube- graphite or a grease type of lube might work. I wouldn't try spray oil like WD40- that will eat up any sort of grease that is inside the cable itself.
Running without an IAC is not a fix, and the car shouldn't run like that.
Other more insane issues would include damage to the IAC wiring harness that is allowing voltage from another circuit in the same harness to get into the IAC wiring. You might have to find this one the old fashioned way of simply tracing down the wiring.
I'll try to think of some other stuff. Oh, your car does have a MAP or MAF somewhere. EFI won't work without it, but it's not the problem. Those don't control engine idle speed. Another idea might be that there is a leaking fuel injector, but I'd think you'd have codes for the O2 sensor if that was the case. Another way to check is by looking at spark plug deposits, but if you have platinum or iridium plugs, then it takes a very long time for deposits to build up.
I'll try to think of some other stuff, but for now I have to go to work.
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Originally Posted by the user formerly known as ZX3_Chick
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