Q1: in fact, for a skyscraper sheaf $\mathcal{S}$ on a surface one has $c_2(\mathcal{S})=- \mathrm{length}(\mathcal{S})$: this follows for instance from example 15.3.1 in Fulton's *Intersection theory*.

Q2: It depends what you call "easy"... It is a nontrivial computation due to Lübke, *Chernklassen von Hermite-Einstein-Vektorbündeln*, Math. Annalen 260 (1982), 133-142.