I am trying to complete Ophala's artifact sidequest (Original Campaign), and I cannot enter the Hodge or Rumbottom Estates. I am a Diplomat-Paladin, and I cannot find the keys to the estates, nor is there an option to persuade the guard to open the door. Do I get the key during the main quest in the area, or otherwise how do I get into the estates?

Tomi can unlock it for you, or bash the door down. The guards won't care.

Just talk to the guards, they literally tell you they don't care if you go in. Either bash the doors, unlock them if you're a rogue or use Tomi as a henchmen and get him to do it.