GATE CSE 2009, Which of the following is FALSE?
A is true, but I don't find your argument convincing (it simply restates the claim). See this question for example.
B is not the correct answer to this problem, because as you say it's true. Your reasoning isn't very formal though, and swaps some reduction orders. A correct reduction would be NFA → (superset construction)→ DFA → (trivial) → PDA.
C seems fine to me.
D is indeed left over, and is indeed false. Inherently ambiguous context-free languages can be recognized by nondeterministic PDAs, but not by deterministic PDAs.