Well frankly, in any real murder mystery the investigators are only in danger when they get close to a solution, right? If the culprit is among us it would be silly for them to start killing us all off one by one, especially if we were headed down the wrong path. So in that case the culprit would want to deceive and misguide the rest of the team in an effort to keep us off the trail without drawing our attention back to him/her. As such the single character/high risk endevor isn't too much a concern until we start getting too close to the truth/the end of the role play. Early round deaths should be reserved for NPCs so that the GM can build tension and suspicion between the team members. It doesn't need to be hard at all to decide. You are the GM, you make the rules. Tell the culprit that they can kill only after x rounds, or send them a PM when they are allowed to, or if you feel it is too early, let them attempt it and get caught at it and end the game because we know who-dun-it.

As to the together by chance sort of situation, there would already be some of that suspicion/tension built into it. If these guys think they are all vigilantes that were so eager to paw over a cadaver, then that might make me be on my toes a bit, watching out more for a back stab. If I am brought into a room with a bunch of other players and am told, hey prove your innocence by finding the real killer among you, if any. I am not going to trust any of the lot, and every single thing anyone says or does will be put into suspicion. 

But hey, I am flexible, and eager to get started!