To grab the whole alphabet
\documentclass{article}
\DeclareMathAlphabet{\altmathcal}{OMS}{cmsy}{m}{n}
\usepackage{mathptmx}
\begin{document}
$\altmathcal{F}\mathcal{F}$
$\altmathcal{S}\mathcal{S}$
abc\fontfamily{cmr}\selectfont abc
\end{document}

To grab just one or two characters
You could save them each in a box before loading mathptmx. No need to worry about smaller mathstyles, as \mathcal does not seem to support them anyhow.
\documentclass{article}
\newsavebox\mcFcontent
\savebox\mcFcontent{$\mathcal{F}$}
\newcommand\mcF{\usebox{\mcFcontent}}
\newsavebox\mcScontent
\savebox\mcScontent{$\mathcal{S}$}
\newcommand\mcS{\usebox{\mcScontent}}
\usepackage{mathptmx}
\begin{document}
$\mcF\mathcal{F}$
$\mcS\mathcal{S}$
\end{document}

One can make it a macro \savemathcal{} and use them with \altmathcal{}:
\documentclass{article}
\newcommand\savemathcal[1]{%
\expandafter\newsavebox\csname mc#1content\endcsname%
\expandafter\savebox\csname mc#1content\endcsname{$\mathcal{#1}$}%
\expandafter\newcommand\csname mc#1\endcsname{%
\expandafter\usebox\expandafter{\csname mc#1content\endcsname}}%
}
\newcommand\altmathcal[1]{\csname mc#1\endcsname}
\savemathcal{F}
\savemathcal{S}
\usepackage{mathptmx}
\begin{document}
$\altmathcal{F}\mathcal{F}$
$\altmathcal{S}\mathcal{S}$
\end{document}
As to your second question, https://ctan.org/search/?phrase=mathptmx says
Package mathptmx
Use Times as default text font, and provide maths support
newtxtext,newtxmathpackages, based onTeX Gyre Termes, a Times clone, which has more possibilities. – Bernard Sep 14 '17 at 18:58