I hate to get murder jobs in the game (what should I tell Michelle?) Until now I basically played the missions as usual, but as the game is "open-world" (well, is it?) I thought I could maybe play it differently. Right now I'm right before the "No Love Lost" mission, where I have to kill the biker boyfriend of Faustin's daughter. I don't mind getting unnerving people out of my way, but I'm certainly not born to be a hitman for other assholes. How can I avoid killing people in the game but still advance in missions?
It's... not that sort of game. Some of the missions explicitly require you to kill one or more people - including unarmed people. It's disconcerting, yes, but I'm pretty sure there's no way to advance the plot without doing these kinds of missions. The GTA series is infamous for the terrible things you do - it has missions in which you beat up juries, missions when you distribute drugs to kids, missions when you bury people alive, etc.
Alternatively, you can forget about the main plot entirely and just play the game as an open-ended sandbox, taking jobs as paramedic, vigilante, competing in races, etc. (though you have to kill some people just to unlock those options...).
One final thing, if you want to avoid running over pedestrians during intense car chases, use the horn! Pedestrians will run away and jump out of the way when you honk. It's also useful for warning other cars - so useful, in fact, that I've bound the horn to one of my mouse buttons (playing on the PC).
IIRC, for the "Kill Faustin's Kid's Inappropriate Boyfriend" mission, you can convince him to leave without killing him. In a lot of "kill people" missions, you have the opportunity to just run them off.
However he's a hardcore scumbag, and armed to boot. I wouldn't worry about killing him. Save your worry for when you have to snuff more ethical people.