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. Unfortunately it is more complicated than that @George, I think you need to add an entry to As
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.