The problem is that I haven't found the list of symptoms that would indicate that the secondary air pump can causes rough running. The car should run fine with or without it. Also it doesn't make sence that starting it up with 5C coolant temp everything is fine, but at 0C big problems appear. If you could link me information where it is said that SAP is causing rough running troubles around 0C I would really appreciate it.

For me the secondary air system has one job to do:
During cold start, the engine efficiency is not the best so there will be more hydrocarbonates going to the exhaust. Blowing more air to the exhaust will rise the exhaust gas temperature and help heating up oxygen sensors and catalytic converter quicker so that the car could go to closed loop where it can work more efficiently.

Having this though in mind, I don't see how secondary air could change the way engine is running while cold. Basically the secondary air will never enter the engine and doesn't participate in the cylinder filling procedure, it will enter the exhaust ports only.

Yes, I do have the codes and it is a very common problem with these engines, but it is very expensive to fix and I haven't found a reason to do that. People who are fixing it are usually from USA where the car is checked for fault codes during the yearly inspection and if they have this code, they will fail at the inspection. In europe the CEL light is not illuminated and the codes are not checked during the yearly inspection, at least in my country.

My current guess is that it is software related problem. But you never know, some have had Vanos seal issues with same symptoms too.