Does local cohomology commutes with direct sums?

Convince yourself first that $H_I^n(M) = varinjlim_k operatorname{Ext}_R^n(R / I^k, M);$ then, use the fact that Ext commutes with finite direct sums in the second component, i.e., $$operatorname{Ext}_R^n(R / I^k, oplus_{i = 1}^m M_i) cong oplus_{i = 1}^m operatorname{Ext}_R^n(R / I^k, M_i).$$

For the first fact, use the definition of the local cohomology modules as the right-derived functors of $Gamma_I(M).$ Convince yourself that $Gamma_I(M) cong varinjlim_k operatorname{Hom}_R(R / I^k, M);$ then, use the facts that (1.) direct limits commute with cohomology and (2.) Ext is the right-derived functor of Hom.

Unfortunately, I am not aware of a more straightforward proof than this.

Yes because homology commutes with direct sums. Alternatively you could use the formulation $$H_{mathfrak{a}}^{n}(-)simeq varinjlim_{t}text{Ext}_{R}^{n}(R/mathfrak{a}^{t},-)$$ combined with the fact that $R/mathfrak{a}^{t}$ is finitely generated to show that local cohomology commutes with all direct limits; in particular it will commute with direct sums.

Edit:

Since $R/mathfrak{a}^{t}$ is finitely generated, there are isomorphisms $$text{Ext}_{R}^{n}(R/mathfrak{a}^{t},varinjlim_{J}N_{j})simeq varinjlim_{J}text{Ext}_{R}^{n}(R/mathfrak{a}^{t},N_{j})$$ for any directed system ${N_{j}}_{J}$ of modules and $ngeq 0$. Consequently one has isomorphisms $$begin{align*} H_{mathfrak{a}}^{n}(varinjlim_{J}N_{j})&simeq varinjlim_{t}text{Ext}_{R}^{n}(R/mathfrak{a}^{t},varinjlim_{J}N_{j}) \ &simeq varinjlim_{t} varinjlim_{J}text{Ext}_{R}^{n}(R/mathfrak{a}^{t},N_{j}) \ &simeq varinjlim_{J} varinjlim_{t} text{Ext}_{R}^{n}(R/mathfrak{a}^{t},N_{j}) \ &simeq varinjlim_{J} H_{mathfrak{a}}^{n}(N_{j}) end{align*}$$ for every directed system and $ngeq 0$.