Allow downspout discharge to cross a sidewalk in a freezing climate?
The preferred solution that I've seen is to discharge the gutter through a couple inch air gap into a catch basin that outputs into the underground drain. The air gap allows for water to drain into the yard when the catch basin freezes solid.

And while it's a bit late to change your underground drains, I prefer the solid pipe installed with a slight slope over the corrugated tubing. It's less likely to collapse over time and water doesn't have anywhere to be trapped in the tube.