I don't know what plot I'm supposed to make in problem 1.
The horizontal axis is cos theta(zenith), and the vertical axis is the number of nu_mu which survived oscillation relative to the original number of nu_mu. This is supposed to be a histogram with 5 bins. Therefore, each bin contains an integral of the neutrino oscillation probability over a certain range of cos theta.
When we check that the probabilities are real in 2(2), I assume that you are talking about the formula (2) with three generations?
Yes. Once you specialize the formula to two generations, the probability is manifestly real and there isn't anything to check.