The doctrinal dispute resulted in the dismissal of the

recently discussed but OA is A, not D

In B: if we don't use with, we would have ambiguity about whether misconduct was parallel with teaching or with doctrine. In other words, it sounds like the president was teaching administrative misconduct!!

In D: The key here is to notice that doctrine is the object of teaching and should therefore be kept as the object, not as an adjective (or, as we sometimes say, "in attribution").

In other words, we should follow this pattern: PREP + GERUND + OBJECT OF GERUND

Example will make this rule clearer - Consider the following sentences:

The woman was fired for buying junk.
or
The woman was fired for junk buying.

After eating lunch, I felt sleepy.
or
After lunch eating, I felt sleepy.

A is my pick

B, C & E are wrong :since he wasn't teaching both-doctrine and administrative misconduct

D a close second pick

Having the same error twice gets me really nervous. It makes me feel i am not learning anything

The key here is to notice that doctrine is the object of teaching and should therefore be kept as the object, not as an adjective.

I think the key point is that sometimes we didn't know doctrine should be object or adjective, both of which are grammatical.
Therefore, we make the same mistake twice.