Hi,
0W20 is better option for very cold winter because it doesn't get ticker while cold. 5W20 is also a good option for winter and summer. Synthetic oil is superior to conventional in all aspects and last longer. Maybe, you can request synthetic oil change from dealer and pay a bit more if you plan to keep your car for long time.

 

Read your car's User Manual to see what oil viscosities are acceptable.
Almost all 0w-20 oils are synthetic.
You shouldn't worry about using 5w-20 vs 0w-20 but synth or synth blend would be better choices.
Sometimes oil bottle may say Conventional oil while in fact it's a synth blend.

 

I would not switch to conventional, I find it hard to believe the dealer would do that. Weight isn't nearly as important, providing you're not too far off. Being in SoCal, weight makes hardly any difference at any weight for us.

 

Do not switch from Synthetic to Conventional, it will destroy the engine. The manual and the cap say to use 0W-20 for all climate regions. Don't deviate.