Mazda 3 start problem
 

I have a 2005 Mazda 3 and have had a lot of problems starting it over the past couple of years. At first the problem only occurred when it was cold out (near freezing). The accessories would all turn on but the car couldn't start. After about 45min though of sitting there, it strangely would, and then act like nothing happened. In previous years when I would bring it to the dealership they would replace the starter relay and maybe the battery. It would work again for about a week, and then not work any more. This year though I brought it to a new dealership and they said it was definitely the immobilizer based on the flashing key and first replaced the key (which didn't fix the problem) and then wanted to replace the immobilizer system and the pcm. I figured that was a bit much, and didn't want to just blindly throw cash at the problem hoping in the process we would fix it.

The next day the car wouldn't start again, but out of curiosity I put a space heater in the car and pointed it at the key hole. After a couple mins the car started up perfectly. This worked great for about a week, but then the starts seemed to be a little slow, and now it doesn't start at all regardless of temperature. Instead, when I turn the key to start, the dash lights (e.g. park, parking brake,oil,check engine...) all start flickering, and there's a clicking noise. When i turn the key back off all the needles (fuel, temp,speed...) all spike in unison and then go back to 0. Also, with the key in "on" pressing the brake momentarily dims all the dash lights and turns off the radio. There doesn't seems to be any problems in the accessory position. Thanks for any suggests you all might have!