The sections of the kernel of $\mathcal{O}_C \to (i_{C_n})_*\mathcal{O}_{C_n} \to 0$ are functions on $C$ vanishing on $C_n\cap D=\Delta$, this gives you a function on $D$ vanishing on $C_n$, hence a section of $\mathcal{O}_D(-\Delta)$, you can check that this defines actually an isomorphism of sheaves. For the second question: it depends on what you know about the curves. What I had in mind was the sequence you wrote.

In your notation $\mathcal{I}(C_n)=\mathcal{O}_D(-\Delta)$ where $\Delta=C_n\cap D$, then you should be able to say something about those cohomologies. Working in this way you can prove a famous formula by Hironaka which computes the arithmetic genus of a reducible curve, see the answear by Georges Elencwajg in this post math.stackexchange.com/questions/1297433/…. Using this one you can argue by induction on the number of components, supposing as you do $C=C_n\cup D$.