The Navy is understandably cagey about how they surface through ice-sheets, but this Smarter Every Day video is the best summary I have seen. https://youtu.be/XFJnWp1tAdU

Basically, some part of the sub needs to be strong enough to reliably break ice at sub-zero temps. The rest needs to maintain its strength at those temps.

Yes, U.S. Navy nuclear submarines go under the polar ice caps somewhat frequently -- as did Soviet (and do Russian) nuclear submarines as well. It's a good place to hide from prying eyes, as not only can ships not (easily) follow you there, but the sound produced by all the ice moving and grinding makes passive sonar quite ineffective.

The YouTube channel Smarter Every Day recently had a video series on a nuclear submarine under the arctic ice.