(a) Your graph is correct, but it would help if you identified the three sets of vertices. They should be grouped as in the diagram below:
1 1 1 2 3 3 2
(b) Trivial.
(c) Let the vertex classes of $K_{\ell,m,n}$ be $V_\ell,V_m$, and $V_n$, containing $\ell,m$, and $n$ vertices, respectively. Each vertex in $V_\ell$ is joined by an edge to each vertex in $V_m\cup V_n$, so each vertex in $V_\ell$ has degree $m+n$, and the sum of the degrees of the vertices in $V_\ell$ is $\ell(m+n)$. In similar fashion you can calculate the sums of the degrees of the vertices in $V_m$ and $V_n$ and add them to get the sum of the degrees of all of the vertices in $K_{\ell,m,n}$; then use the handshaking lemma to find the number of edges.
(d, e) A graph has an Euler circuit (or trail) if and only if the degrees of its vertices satisfy a certain condition, and in (c) we saw how to calculate the degrees of the vertices; just combine this information properly.