Mfcttrf

Let $X$ be a (finite if you want) set and form the simplicial set $F^bullet(X)$ with
$$
F^n(X) = mathrmHom_mathrmset ([n], X)
$$
where the right hand side denotes arbitrary maps of sets (of course
it wouldn't make sense to say order preserving as $X$ doesn't come
with an order).

I'm wondering about a description of $F^bullet(X)$. For example if $X = 0,1$ then there are 2 0-simplices, may as well call them $[0] and [1]$ and 2 1-simplices $[0, 1]$ and $[1,0]$ glued together to form a copy of $S^1$.

Edit: as pointed out in Goodwillie's answer, this is not the end of the story, there are way more higher dimensional non-degenerate simplices.

Is there an analogous description when $X = 0, 1, 2$?

A closely related question is whether there's a right adjoint to the
forgetful functor from the simplex category $Delta$ (finite ordered
sets) to, say, finite (unordered) sets -- and if so what is it.

Example where such simplicial sets arise: given a map of topological spaces $f: X
to Y$ we can always form a
simplicial object $mathcalS^bullet(f)$ with
$$
mathcalS^n = prodnolimits_X^n = underbraceX times_Y
cdots times_Y X_ntext times
$$
with face and degeneracy maps given by projections and diagonals
respectively. Taking connected components gives a simplicial set.

When $Y$ is the union $bigcup_i=1^N H_i$ of the coordinate
hyperplanes in $mathbbC^N$ and $f: X=coprod_i=1^N H_i to
bigcup_i=1^N H_i=Y$ is the obvious map, I believe the simplicial
set we get is $F^bullet(1, dots, n)$.

You are overlooking some nondegenerate simplices. For example, when $X=0,1$ there are the $2$-cells $[0,1,0]$ and $[1,0,1]$. In fact, the thing you call $F^bullet(X)$ is infinite dimensional if $X$ has more than one element.

It is contractible whenever $X$ is non-empty; this can be seen by identifying it with the nerve of a category, a category equivalent to the point category with one morphism.

If $X=G$ has a group structure then $F^bullet(G)$ is often called $EG$; it is a contractible space with free $G$-action.