Question: Can I use 15w40 synthetic oil in my 2015 vw jetta tdi and how long can go between oil changes?

Answer: 
Replace the battery and those issues will stop.

Reason? The bad battery is so discharged after starting the engine the alternator takes a few seconds to bring the system voltage high enough for some electronics to work properly.
The same belt is possibly driving the power steering pump and the alternator. If the belt is slipping when you first start the engine, then the power steering will not work correctly and the alternator will not be providing power.

Friction on the slipping belt may then make it heat up and start to grip. The power steering would then start operating and the alternator start producing power. The increase in voltage when the alternator starts working would make the AC fans run faster.

You may need to re-tension the belt, or replace it. 

If this car has an electric power steering pump, then alternator belt could still be slipping and would therefore not be able to supply enough power to the electric pump and the AC fans until the belt stared to grip.

Either way, I would check the belt before replacing the battery.