Thanks for information, guys. I found where the fuse is located (panel on the lower left drivers side), though decided not to restart the system this time.
I brought the car to the dealership service, and they found that it was an issue with the climate control software produced by Microsoft, but not with Sync (I have all latest updates for Sync). I was told that Ford is familiar with the problem and that they already released an update that supposed to fix this issue. So, guys in the service just installed that update and returned me a car. Thankfully, all of this took half a day and I was given a rental car. So far so good, everything works. Hopefully, this will not happen again...