I don't know about terminology, but let me try to formalize what I think you mean, in near-maximum generality. Perhaps this is really what you're looking for; if not, oh well.
This answer assumes a basic understanding of von Neumann ordinals. The second part assumes a basic understanding of functors (in general), and the product functor in particular.
Definition 0. Suppose $X$ is a set and $\alpha$ is an $n$-long sequence of ordinals. Then an $X$-valued array of shape $\alpha$ is a function $X \leftarrow \prod_{i<n} \alpha_i.$
So for example, if $A$ is an $X$-valued array of shape $(3,4,5)$, then the value stored at $(1,2,1)$ is denoted $A(1,2,1)$, and this is an element of $X$.
Anyway, since arrays are just functions, we can use standard mathematical notation to get your "slices". For example, if $A$ has shape $(3,4,5)$, then $A(1,-,-)$ is the corresponding array of shape $(4,5)$ defined in the obvious way. So the two-dimensional "slices" obtained by choosing different values for the first index are
$$A(0,-,-) \qquad A(1,-,-) \qquad A(2,-,-)$$
and these are all $X$-valued arrays of shape $(4,5).$
Your second question is much more interesting. The amazing thing about ordinals is that the following definition makes sense:
Definition 1. Suppose $\beta$ is an ordinal and $S$ is a subset of $\beta$. Then we write $\mathrm{ord}_S$ for the unique ordinal that is order-isomorphic to $S$, and we write $S^*$ for the unique such isomorphism $S \leftarrow \mathrm{ord}_S$.
The cool thing about this is the following. Suppose we're given an array $A$ of shape $(10,10,10).$ Suppose I'm interested in the "subarray" given by [3->4, 1, 4->9]. Well, this can be formally defined as:
$$A \circ ([3,4]^* \times [1]^* \times [4,9]^*)$$
where $\times$ is understood to be a functor in the usual way.
