Any graph $G$ can be seen as the sum of complete $k_i$-partite graphs?
I think the answer to your first question is "no".

In particular, if you take a random bipartite graph $G=([n],[n],E)$
where $Pr[(i,j)\in E] = 1/2$ for each pair $(i,j)$,
the answer is no with probability close to 1.

I am interpreting your first question as follow:

Given any graph $G$, is it always possible to find a collection of subgraphs such that 


each subgraph is a complete $k$-partite graph for some $k$,
each edge in $G$ occurs in at least one of the subgraphs, and
each vertex in $G$ occurs in at most a constant number of the subgraphs?


Proof.
Fix $n$.  Let $G$ be the random graph described above.

Since $G$ is bipartite any complete $k$-partite subgraph has to be bipartite.

First, I claim that with probability $1-o(1)$, every complete bipartite subgraph of $G$ has at most $4n$ edges.  

(Here's why.
For any pair of subsets $L\subseteq [n]$ and $R\subseteq [n]$ with $|L\times R| \ge 4n$,
the probability that the complete bipartite subgraph with edge set $L\times R$
is in $G$ is $2^{-|L\times R|} \le 2^{-4n}$.
There are fewer than $2^n\times 2^n = 4^n$ such pairs $L$ and $R$,
Thus, by the naive union bound, the probability that any of
the corresponding subgraphs is present in $G$ is at most $4^n 2^{-4n} \le 2^{-2n}$.)

Also, with probability $1-o(1)$, the graph $G$ has at least $n^2/4$ edges.

Thus, with probability $1-o(1)$, every complete bipartite subgraph in $G$
has at most $4n$ edges, and $G$ has at least $n^2/4$ edges.
Assume this happens.  

Now suppose for contradiction that a collection of subgraphs with the desired properties exists.
Each of the subgraphs has edge set $L\times R$ for some pair of subsets $L$ and $R$.
In $G$, direct all the edges in $L\times R$ from the larger side to the smaller side
(to the left if $|L|\le |R|$, and to the right otherwise).
Since $|L\times R|\le 4n$, the smaller of $L$ or $R$ must have size at most $2\sqrt n$.

Since each vertex is in $O(1)$ of the subgraphs, each vertex now has $O(\sqrt n)$
edges directed out of it.  But all edges are directed one way or the other,
so the number of edges in $G$ is at most the number of vertices 
times the maximum out-degree of any vertex, that is, at most $O(n\sqrt n)$.
This contradicts the graph having at least $n^2/4$ edges.