IMMOBILIZER problem in winter - cold weather

I got a BIG PROBLEM with starting in winter. I own MAZDA 3 (year 2004), I loves it to bits. But when temperatures drop below 0 (basically in winter only) my car does not want start.
Everyone was telling me it must be battery. But I was not sure, because it even wont try turn engine. But I got new battery and it did not solve the problem. My friends checked electric parts and it looks fine.
Then in MAZDA service told me to check IMMOBILIZER CONTROL if it is always on, or if it turns off when I get key in first position. Found out in cold weather it is always on and won’t let me start. So it is immobilizer problem we think (I am confused anyway).
But when I cannot start I turn on ventilation in car for few minutes to heat up car and my immobilizer let me start after that. It is very strange. Or once I start in the morning, then my car starts ok until next morning day when it cools down everything in the car.
In MAZDA service they checked connectivity on ignition etc. and they sprayed connectors for good connectivity. But if it wont solve my problem until tomorrow morning what do you advice?

Did anybody have same problem ? I hate winter. I am worried go anywhere that I wont start car

 
Suggest moving to a warmer climate.  Take the Mazda with you.  
 
Nah Man. That's a new one on me. It's certainly not symptomatic of all Mazda3's, but probably a first!
 
But if the car won't start, how does using the heat/defrost/AC fan warm the car?  That's the only thing you've said about this problem that doesn't make sense to me.  It can only get warm if the coolant temp is up.  And that can't happen unless the engine has been running.
 
 

Well I know what you mean and I know it sounds strange. But I think when outside is lets say -5 and I will turn on only fan without engine on, bit warmer air is goin throught then -5. So immobilizer will get little bit warmer.
Or maybe it is only my thinking that it helped. But it is very strange for me anyway. And I have no idea how to solve this problem. Tried it tonight (after they cleaned the contacts) and firstly it wont let me start and second try it was ok. Mysterious for me.
Befor they cleaned it in Mazda service it wont let me start unles I was playing with key (taking out from ignition for maybe 15 times, switching A/C fan on etc.).

There is nothing wrong that car what cannot start !!! I will try it morning again :(

 
DAYEM!!! -5*!!  Wind too?  You'd BETTER move to a warmer climate.  Your best choice.
 
Say you turn on the headlights, or radio (or whatever other electrical device) for the same amount of time as you are the fan now.  Will the car start then?
 
 

Yep, at the moment it is -7 lol. At the moment it is not windy but few days ago was very windy lol. Poor my little Mazda 3.
Well, you know when your battery has not got good capacity and your car wont start because of battery. Advice for that is you flash with your headlights. Then battery will get shock and might start.
But I was doing then anyway in my case. If it helped I do not know, because my problem is in immobilizer. But it might warm contacts on immobilizer I think.
I think I will be taking heater with me to the car lol. I was planning to go to the mountains with my car, but hey what if it want start ? I will be stucked in mountines forever lol !
I am thinking go to Mazda again and ask them disconnect immobilizer forever. But do not know if it is so easy to do that ... I believe they just connect laptop to the car and press off button lol