Ok, I think I figured out the Inga part of Case 5. Just put in some conjecture, and you can explain away a lot of the possible plot holes.
Several days before the start of the case, Inga kidnapped Maya. Dhurke learns of this, and goes to the Tomb to try and save her. Inga shoots him 3 times, and Dhurke pretends to be unharmed. This scares Inga away, and we must assume that due to his inability to see faces, does not know if it was the real Dhurke he shot.
Dhurke quickly dies because bullets kill people. With his final words, he instructs Maya to go back to the United States, so he could see Apollo one last time. Maya goes back to the United States, changes into a Dhurke outfit, and begins channeling Dhurke.
At this point, Dhurke's body is simply on the floor of the Royal Tomb, but doesn't immediately get found. Ga'ran learns of this though her Royal Guard. Ga'ran also knows about Inga's plans to get the founders Orb and do a coup. Since she wants to recover the Orb herself, she comes up with the plan to kill off Inga, and frame Dhurke for the crime, to have the triple whammy of getting rid of Inga, getting the orb, and help crush the rebels spirits. She hides Dhurke's body in Amara's tomb, and makes Inga believe that Maya is still in the tomb by having a guard dress up like her, along with believing Dhurke was not killed.
She soon infers what happens to Maya, when she find out Dhurke is supposedly alive in the United States. At this point she reaches to Amara for help to pulling off the full plan to frame Dhurke of Inga's murder.
Ga'ran at 2 sneaks into the tomb in Dhurke's outfit, and and stab's Inga. She then puts Dhurke's outfit back on Dhurke, unties the guard she left there, removes the Maya outfit from him, and leaves the tomb as if nothing happens. At 2:30 Amara channels Inga's Spirit in his room, dressed up in his outfit, with the expectation that Inga would simply believe he had a nightmare and rush back the tomb for the hostage exchange. This proved to be the correct assumption, and once he reached the tomb, Amara stopped channeling Inga and burned the outfit. At 2:45 when Maya channeling Dhurke came in, Amara hit her with the red magatama, to force the spirit to leave Maya. She then dressed her back up in her outfit, and tied her the chair. After that, she dressed as as Dhurke, made it look like he killed Inga, and waited until she heard the others come. Once she did, she quickly channeled Dhurke, to help pull off the framing.