It is a fact (following from the Ehrenfeucht–Mostowski theorem) that for every complete theory $T$ and for every $\lambda \geq |T|$, there is $M \models T$ with $|M| = \lambda$ and $M$ having $2^\lambda$-many automorphisms (assuming $T$ has infinite models). So if I'm understanding your question correctly then what you denote has $\mbox{Mod}_\kappa(T)$ always has cardinality $2^{<\kappa}$, at least for $\kappa > |T|$.
PS: To partially address your second question: for every cardinal $\lambda$ (possibly $\lambda$ finite or $0$) there are $M \equiv N$ structures in a countable language such that there are exactly $\lambda$ elementary embeddings from $M$ to $N$. Indeed let $T$ be the complete theory of $(\mathbb{Z}, <, 0)$. For $\lambda = 0$ note that there are no elementary embeddings from $2 \times \mathbb{Z}$ to $\mathbb{Z}$, and for all other $\lambda$ there are exactly $\lambda$ embeddings from $\mathbb{Z}$ to $\lambda \times \mathbb{Z}$.