The Key to Kharag's Room can be pickpocketed off Kharag gro-Shurkul, but it does not unlock anything. When entering the Solitude Sawmill building, it is found that Kharag's room has no door on it, and thus, the Dragonborn cannot use the key to unlock it.
Kharag gro-Shurkul may send hired thugs after the Dragonborn if they steal from him.
He will not let the Dragonborn cut their own sawn logs. It is unknown if this is a bug, developer oversight, or intentional.