Agree with you about trade and exploration not working during a war/civil war, salvage and data also don't appear to have an effect even though the missions are described as helping the war effort, but I think assassination missions still count.Only three things can affect influence during a Civil War. Killing ships which has negative influence on the faction they're aligned with. Turning in Combat Vouchers from CZ's which has a positive influence on the faction that issued them. And completing combat missions for a faction which has a positive influence on the faction. All form of trade, exploration, and other mission types will have no effect on influence.
I have no idea about passenger missions. I wouldn't be surprised if some passenger types still had an effect during a war.