Why don't we see a $\frac{\times}{ \div}$ like we see $\pm$? It's common to see a plus-minus ($\pm$), for example in describing error
$$
t=72 \pm 3
$$
or in the quadratic formula 
$$
x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}
$$
or identities like
$$
\sin(A \pm B) = \sin(A) \cos(B) \pm \cos(A) \sin(B)
$$
I've never seen an analogous version combining multiplication with division, something like $\frac{\times}{\div}$

Does this ever come up, and if not why?

I suspect it simply isn't as naturally useful as $\pm$.  
 A: Perhaps because 
$$
a\frac{\times}{\div}b
$$
(typographically quite horrible) is written as
$$
a\cdot b^{\pm1}
$$
A: As you indicated,  square root can be + or -.
$\pm$ shows this ambiguity.
As far as I know,
there is no similar use case
where the choice is
to multiply or divide
by an expression. 
A: I think that this question is primarily opinion-based, so here is my opinion:


*

*The expression $[t=72\pm3]$ is equivalent to $[t=72+(+3)]\vee[t=72+(-3)]$

*The expression $[t=72\frac{\times}{\div}3]$ would be equivalent to $[t=72\times3]\vee[t=72\times\frac13]$


So the second operand "looks the same" in the case of $\pm$ but not in the case of $\frac{\times}{\div}$.
If we had a different notation for $\frac13$ (for example, $\color\red3$), then it might have seemed more appropriate to denote something like $[t=72\frac{\times}{\div}3]$, which would be equivalent to $[t=72\times3]\vee[t=72\times\color\red3]$.
So it's basically a matter of "backward compatibility" with our existing notation for inverse...
A: There are many function (square roots for ex.) where $f(x) = f(-x)$ and for $f^{-1}$, the $\pm$ is useful. If there were common functions where $f(x) = f(\frac{1}{x})$ it might be a thing. Can anyone think of example functions like this? 
A: The multiplication sign $\cdot$ is usually omited in abelian groups (or even non-abelian). I think the most important thing is, that something like $\div$ does mislead the reader, because if we consider $a\div b$, then $b$ is not on the same level as $a$, but in contrary the sumbology does suggest so. This can cause serious errors in calculations whereas $\frac{a}{b}$ is much more clearer as well as $ab^{-1}$. I think this levelness is the main point of never using $\div$.
