Why lowercase letter are not allowed in \mathcal ?

To be honest, I am not sure for the reasons behind this.

There are some related posts on TeX - LaTeX, but I did not notice some of them mentioning why this is the case.

For example: Lowercase \mathcal, Why do lower case mathcal letters show up as random symbols? or How to get the lowercase calligraphic symbols?

Considering your last link, we need some user packs for making scripted lower case letters.

Yes, it seems that there are some ways how to do that in LaTeX. However, I am not sure whether some of the solutions are transferable to MathJax.

OTOH, in MathJax there is no problem with using Unicode characters.