I have been struggling on this puzzle, is it possible for Humanities to achieve the goal?  Is there a way to find holes in Argos' contract with only 3 questions? 

Hello, Pocket! Yes, with the exception of Speedrunner (who doesn't even get the opportunity to question Argos) all backgrounds can find loopholes in the his statemets, three questions are more than enough (although the Leader can negotiate for five). All I will say is this: read all the terms carefully: one of them is faulty and easy to exploit to your benefit.