As for synoptic scale weather, I don't think a tropopause fold affects the synoptic scale weather events much, but are rather a part of the weather events. The downward mixing of stratospheric air will introduce drier air, leading to lower humidity in the general vicinity. Of course, ozone will be entrained into the tropopause as well, which has air quality implications. And there are certainly feedbacks in the fold phenomena that will affect heat, radiation, and kinetic patterns... but the synoptic weather in the region is going to be more driven by the jet stream itself rather than a fold that may occur. A meteorologist, though, might have better information on that.