Is there a necessary form of consecutive composites? For every $n \geq 3$ there is a tuple of $n-1$ consecutive composites, namely the composites of the form $n! + 2, \dots, n!+n$. However, must a tuple of $n$ consecutive composites take the form? It seems plausible to me, and I have not yet seen a counterexample.

No, we can find consecutive composites that are not of this form. The point of $n!$ is just that it is a "very divisible number". We can obtain a lost of many other examples just like this one.
For example the numbers $n!^2+2,n!^2+4\dots +n!^2+n$ or $n!^3+2,n!^3+3\dots n!^3+2$.
Also $kn!+2,kn!+3\dots k!n+n$ works for all $k>0\in\mathbb Z$
You can also get a smaller examples if instead of using $n!$ we use the least common multiple of the numbers between $1$ and $n$.
