I didn't get the impression that they were being "ordered" to kill Gavilar. Gavilar told them what his plans were, the five Parshendi realized that what he was doing was going to cause their gods to return, it freaked them out, so they made a desperate move to kill Gavilar, even though they knew that the humans would probably wipe them out in revenge. But they felt that them getting wiped out by humans was a better situation than having to deal with their own gods. I don't think they worship their gods. I think they're terrified of them.
My guess, with no evidence, is that the gods of the Parshendi are voidbringers/whatever those 10 dudes are that I've guessed are like the Heralds of the voidbringers, but are sort of Lovecraft-sounding. I can't remember their name. Not gods in the sense of them being revered by the Parshendi, but gods in the sense that they were powerful, supernatural beings who ruled/dominated in the ancient history of the Parshendi.
I personally don't think the Parshendi are tools of Odium. Dalinar remarks that they don't seem to fight out of rage or bloodlust, they mourn over their fallen and fight united. It's my opinion that the Alethi are the current unwitting tools of Odium, since they are the ones that have embraced the Thrill.
And as others have said, yes, gem hearts for soulcasting food.