Finding a graph where the edge set satisfies a set of integer equations
You are asking two questions.  I will answer the first one.  Your first problem is NP-hard, by reduction from exactly 1-in-3 positive 3-SAT.

Let $\varphi$ be a positive 3-CNF formula with $m$ clauses and at most $n^2$ variables $x_i$.  Label each edge $(v_i,v_j)$ with a distinct variable $x_{ij}$.  Each set $S_i$ will correspond to a clause of $\varphi$ as follows: the set $S_i$ contains one edge $(v_i,v_j)$ for each variable $x_{ij}$ that appears in that clause.  Let $k_1=k_2=\cdots = k_m=1$.

Now the resulting system of linear equations has a solution if and only if $\varphi$ has an assignment where every clause has exactly one of its three variables set to true.