Destroyer gunfire against a capital ship

Did destroyers ever pick gunfights with capital ships, and could they penetrate their armor?

Yes, but ordinary destroyer guns were very unlikely to defeat battleship belt or deck armor.

HMS Somali did choose to engage an Italian battleship with her destroyer guns. Her crew even fired improvised baked-bean cans as part of the attack. Ordinary destroyer shells were very unlikely to penetrate a battleship's belt or deck except under extraordinary conditions. Harassment, distraction, and damage to exposed fittings were therefore more realistic goals than a killing blow against the armored citadel.