2) Colimits for cofinal subcategories are the same. See Mac Lane, Categories for the working mathematician, section IX.3.
1) This is more subtle. What should be an element of an object $X$? In the spirit of the Yoneda Lemma, it makes sense to regard arbitrary morphisms $A \to X$ as "generalized elements" of $X$. Because then the Yoneda Lemma essentially says that an object is determined by its generalized elements, which resembles the extensionality axiom in set theory.
Assume we have a directed colimit $\mathrm{colim}_i X_i$ in a category $C$, and that $A$ is a finitely presentable object of $C$. By definition, this means $\hom(A,\mathrm{colim}_i X_i) = \mathrm{colim}_i \hom(A,X_i)$, so that in fact every generalized element of $\mathrm{colim}_i X_i$ comes from a generalized element from some $X_i$, and that this choice is essentially unique: Two generalized elements of $X_i$ and $X_j$ become equal in the colimit if and only if they become equal in $X_k$ for some $k \geq i,j$. Similar statements hold when the diagram is $\lambda$-directed and $A$ is $\lambda$-presentable for some cardinal $\lambda$.
For example, if $C$ is some algebraic category, then the free object on one generator $F(1)$ of $C$ is finitely presentable (in fact, any object defined by finitely many generators and relations, see Chapter 3 in the book by Adamek and Rosicky on locally presentable categories), which essentially means that the forgetful functor to sets preserves directed colimits. This is not true for arbitrary concrete categories! For example, consider the category of topological vector spaces. The forgetful functor doesn't preserve directed colimits. An element in a colimit of topological vector spaces $V_i$ is just a limit of elements in the $V_i$.
Another example, let $C$ be the category of quasi-coherent sheaves on a concentrated scheme $X$ (for example, noetherian schemes are concentrated). Then $\mathcal{O}_X$ is presentable, which essentially means that the functor of global sections preserves directed colimits. This then can be generalized to its derived functors, i.e. sheaf cohomology, which is quite important and useful.
Best Answer
Denote the direct limit of $\left\{G_\alpha , f_\alpha^\beta \right\}$ as $G$, and the direct limit of $\left\{H_i,g_i^j\right\}$ as $H$. Then we claim that $G\oplus H$ is the direct limit of $\left\{G_\alpha \oplus H_i, f_\alpha^\beta \oplus g_i^j \right\}$. And your questions are answered by this claim.
The morphisms $G_\alpha\oplus H_i\rightarrow G\oplus H$ are induced by the morphisms $G_\alpha\rightarrow G$ and $H_i\rightarrow H$.
Suppose there is a compatible system of morphisms $G_\alpha\oplus H_i\rightarrow M$. Since $\operatorname{Hom}(G_\alpha\oplus H_i,M)\cong\operatorname{Hom}(G_\alpha, M)\oplus\operatorname{Hom}(H_i, M)$, this induces compatible systems of morphisms $G_\alpha\rightarrow M$ and $H_i\rightarrow M$. Then by the universal property of the direct limit, these morphisms factor through $G_\alpha\rightarrow G$ and $H_i\rightarrow H$. This immediately implies that the morphisms $G_\alpha\oplus H_i\rightarrow M$ factor through $G_\alpha\oplus H_i\rightarrow G\oplus H$. Therefore $G\oplus H$ is the direct limit of $\left\{G_\alpha \oplus H_i, f_\alpha^\beta \oplus g_i^j \right\}$ as claimed.
Hope this helps.