Desicision tree classification with a "false" attribute
Maybe I'm not interpreting the scenario correctly, but task 1 seems impossible and task 2 easier than you've made it sound.

I'm reading it as a sample for each patient, with one feature for each student, possible values yes/no/dunno.

In that case, at any split of the decision tree the imposter and the chosen real student will split the data in the same fashion.  If either of them is chosen as the split criteria, then it's random which of the two is used, but if we remove the imposter first, then the real student will be used; but there is not difference in the way the samples are split.

As for finding the imposter, all you can do AFAICT is look for pairs who give exactly opposite responses.  But if there are multiple such pairs, I don't see how to pick one, and even when you have a pair of opposites, I don't see any way to determine which is real and which is imposter.  The 60% figure seems to just indicate that nobody answers "I don't know" too often.  I guess we can make an additional assumption that the students are better-than-random, so that the imposter is worse?