On the threshold problem for Latin boxes

Let $m \leq n \leq k$. An $m \times n \times k$ 0-1 array is a Latin box if it contains exactly $mn$ ones, and has at most one $1$ in each line. As a special case, Latin boxes in which $m = n = k$ are equivalent to Latin squares. Let $\mathcal{M}(m,n,k;p)$ be the distribution on $m \times n \times k$ 0-1 arrays where each entry is $1$ with probability $p$, independently of the other entries. The threshold question for Latin squares asks when $\mathcal{M}(n,n,n;p)$ contains a Latin square with high probability. More generally, when does $\mathcal{M}(m,n,k;p)$ support a Latin box with high probability? Let $\varepsilon>0$. We give an asymptotically tight answer to this question in the special cases where $n=k$ and $m \leq \left(1-\varepsilon \right) n$, and where $n=m$ and $k \geq \left(1+\varepsilon \right) n$. In both cases, the threshold probability is $\Theta \left( \log \left( n \right) / n \right)$. This implies threshold results for Latin rectangles and proper edge-colorings of $K_{n,n}$.