Is is mathematically correct to write $\frac{h>2}{l<4}\neq\frac{1}{2}$ therefore $\sin^{-1}\frac{h>2}{l<4}\neq30^{\circ}$?

The goal is to say that an h>2 dived by an l<4 will never be 1/2 therefore the inverse sine of an h>2 dived by an l<4 can never be 30 degrees. Can I express is like so? $\frac{h>2}{l<4}\neq\frac{1}{2}$ therefore $\sin^{-1}\frac{h>2}{l<4}\neq30^{\circ}$ Or is that mathematically "illegal"?

You would be better to specify the conditions on $h$ and $l$ separately, then use them in the expression required.

For example,

Given $2<h$, $l<4$, $\frac{1}{2}\neq\frac{h}{l}$ therefore $\frac{\pi}{6}\neq\arcsin \left( \frac{h}{l}\right)$

This is because you may be referring to either of the values or terms in your inequality, and using them in different combinations. Do you mean to use the fraction with 2 over $l$ or the fraction with $h$ over 4? It is not clear in the way presented.

Not usually. The inequality "$h>2$" is distinct from the number "$h$." What you're trying to do is have "$h>2$" stand in for "the number $h$, which is greater than $2$," but this isn't common practice, and so it's probably best to avoid it. I would recommend phrasing this in words:

If $h>2$ and $\ell<4$, then $\frac h\ell\neq \frac12$, and so $\sin^{-1}(h/\ell)$ cannot be $30^\circ$.