[Aurel]

Make sure your battery connections are clean. You could have a big ohmic drop that causes your symptoms. The battery voltage will look OK when it is not under load, but will drop too low when you try to crank. That would explain why you lights go dim too.

I had the same problem, and in my case the ohmic drop was at the ignition switch. Note that if this is the case, you could bypass it with a remote starter, as a way to evidence the source of the problem. Since the weather is cold, chances are you just have a weak battery.