This problem tests two topics: IDIOMS and PARALLELISM.

The structure SUCH vector-borne diseases LIKE malaria and rabies is not idiomatically correct. We need to say SUCH... AS. Eliminate A and B.

You're correct that C, D, and E involve parallelism with EITHER / OR. Be careful, though! Just because the correct answer didn't say exactly what you wanted it to say, that doesn't make it wrong. All of the following would be correct:

... have focused on immunizing humans or obliterating --> parallel gerunds
... have focused on immunizing humans or on obliterating --> parallel prepositional phrases
... have focused either on immunizing humans or on obliterating --> parallel prepositional phrases
... have focused on either immunizing humans or obliterating --> parallel gerunds

You're correct that either on X or on Y would have been parallel, but so is on either X or Y.

In PARALLELISM, there is a difference between two-part parallel markers (either/or, not/but, etc) and one-part parallel markers (and, or, etc). For more on how to attack those, see:
https://www.beatthegmat.com/help-manhatt ... tml#713882
https://www.beatthegmat.com/think-of-x-a ... tml#575578

I understand your confusion. In English, words ending in "-ing" can act as either nouns (in which case we call them gerunds) or modifiers (in which case we call them present participles). The context of the sentence determines which case we're looking at.

If we saw "the sleeping baby," then "sleeping" is telling us the type of baby. It does not have to be the case, though, that "immunizing humans" tells us the type of humans. When used with "on," it is understood that "immunizing" is a gerund, not a modifier. There is no significant difference between "immunizing humans" and "the immunization of humans." You or I might prefer the latter stylistically, but the former is not incorrect.

When introducing examples, you're right - the correct idiom is "such as." In this example, the expression was broken up: "such examples as." This is also correct.

We can't introduce examples simply with the word "as." We use "as" to compare clauses:
You like music, as do I.

or to express "in the manner of" or "in acting the part of":
As a teacher, I care about critical thinking.
As a woman, she was offended by the joke.