For ii), write $g(x)=f(x)/(\def\eix{\mathrm e^{\mathrm ix}}\eix-1)$ as $f(x)=g(x)(\eix-1)$. Now if you compute the Fourier coefficients on both sides, the factor $\eix$ shifts the coefficients of $g$ by $1$, yielding $C_n=D_{n-1}-D_n$.
For the first question, you can conclude from $C_n=D_{n-1}-D_n$ that $\sum_{k=j}^nC_k=D_j-D_n$ by induction, then let $j\to-\infty$ and $n\to\infty$ to get $\lim_{n\to\infty}(D_{-n}-D_n)=\sum_{k=-\infty}^\infty C_k=f(0)=0$. Contrary to what I wrote in the first version of this answer, that only establishes that the imaginary part of $D_n$, which is odd, goes to $0$. To conclude that all of $D_n$ goes to $0$, either note that $g$ is continuous at $0$ (and thus everywhere), and conclude that $\sum_{k=-\infty}^\infty D_k=g(0)$, so $D_n$ must go to zero for the series to converge; or invoke the Riemann-Lebesgue lemma, as Siminore suggested in a comment under the question.