Yes if none of the dice match your number, the payoff is -1.
I do not think you understand where these binomial coefficients are coming from / what they represent. The small x represents the number of dice that match your chosen number, not the payoff. So we don't need to worry about what happens if we try to take 3 choose -1.
"Push" means that you get your stake back. Neither win nor lose money.
Is the correct answer for part c of your question a negative number? No, I don't think so.
Given the way that you set up the equations, did you have to get a negative value for c? Yes.