I put the solutions in spoiler text so that you can look at one of them without seeing the others if you want. Mouse over it to see it.
Pacifist: Spare Giselle in chapter 1. This is possible by using Steelsplitters for your attack instead of Tarsiers and holding them back once she appears.
But I Survived: Survive the Photonic Plasma Bombs in chapter 2. This is possible on the final wave, by killing all but one of them you lower the damage enough for it to be possible to survive.
No Thanks: Beat The Warbringer mission without using Amporilla.
No Thanks, Again: Beat Practically Legal without firing the Rainmaker.
I Didn't Do It: Don't kill Rube in the Guardian Rube mission. This is possible by Ragering away your attack on the turn he appears.