The `\mathscr` problem is simply that Org mode LaTeX doesn't by
default load the LaTeX package that provides that command. Just add
`#+LATEX_HEADER: \usepackage{mathrsfs}` to your Org file and that
should fix both preview and PDF export.

As for getting unicode math to work, you need, again, to load a
package (so add `#+LATEX_HEADER: \usepackage{unicode-math}` to your
Org file), but there's a wrinkle this time: the `unicode-math` package
doesn´t work with `pdflatex`, only `xetex` or `luatex`, so you
additionally need to customize the `org-latex-pdf-process` variable (for PDF export) and the `org-preview-latex-process-alist` and `org-preview-latex-default-process` variables (for LaTeX preview) to
choose one of those compilers.