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.

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 originally to Lübke, and extended to the case of Higgs bundles by Simpson (Proposition 3.4 in Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization. J. Amer. Math. Soc. 1 (1988), 867–918).