Let me tell you that both these questions are primary grade Olympiads (grade 2 or grade 3) level questions !
For question-(I), I hope that you will not find it difficult to figure out that options (C) and (D) are wrong. This may be concluded by taking some instances of the given statements, as we do as below:
A) even x even = even ( 6×6 = 36->even) ==> True
B) even x odd = even  (6×3 = 18 -> even) ==> True
C) odd x odd = even (3×3 = 9 -> odd) ==> Wrong
D) even+odd = even (4+7 = 11 -> odd) ==> Wrong
E) odd+odd = even (8+8 = 16 -> even) ==> True
So, it’s easy and it is acceptable for primary grade Olympiad.
Now, for question-(II) the logic goes like this. ==>If ‘n’ is a positive integer then definitely (n+1) and (n+2) are also positive and they are CONSECUTIVE numbers (numbers appearing one after another). As they are consecutive numbers, then one of them is definitely even and the other is odd. So, it turns out that (n+1)x(n+2) = (odd) x (even) or (even) x (odd) = Always EVEN. 
Its given that (n+1)x(n+2) = 20182019==> Which is ODD.
So, the answer should be (D), that is No such ‘n’ exists, as the statement is not possible.
Now, let me tell you that question(II) is a typical South East Asian or Singapore Olympiad, Primary grade question, where the stress is given on application of fundamentals.
So, you can understand that both the questions test same concept but the 2nd question really test where you apply the concept and how you relate the theories learned.
The Singapore math questions compel students to think beyond school syllabus and in turn increases logical bend of mind in true sense.
For more, on how to appear for these Olympiads visit Olympiads-For Champs.
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