The nearby Towers of the Teeth were destroyed in the Downfall of Barad-dûr, but there's no way to be sure whether Durthang met the same fate. If it did somehow survive that cataclysm, though, it would almost certainly have been demolished by the Gondorians soon afterwards.
2
The dark name Durthang would only make sense after Sauron's forces returned to Mordor and captured this castle. Presumably its original builders would have given it a less evil name but, if so, no trace of that earlier name survives.