Welcome to the forum. Problems about mathematical logic should be posted to Discrete Math subforum. In fact, see this post.
What is the scope of the existential quantifier in iii(a): just the premise or the whole implication. Please refer to syntactic conventions about omitting parentheses after the description of the syntax of first-order logic in your textbook. My guess is that the scope is the premise only, i.e., ∃ binds stronger than ->. Then the statement is true. Indeed, is equivalent to x + 1 ≠ 1, i.e., x ≠ 0. In fact, if the scope of ∃ extends until the end of the implication, the formula is still true. Indeed, for x = 0 we can take any y, e.g., y = 1. Then y(x + 1) ≠ y is false, so the whole implication is true. If x ≠ 0, then again we can take any y, and the implication is true in virtue of the truth of the conclusion.
I'll look at iii(b) and (c) later.