New numbering system

Use \value{⟨counter⟩} for obtaining the value of a LaTeX-counter in terms of arabic numerals.

By the way: I suggest removing spaces from the output of the macro completely. So the user can insert them explicitly wherever desired.

\documentclass{article}
\usepackage{amsmath}
\usepackage{etoolbox}

\makeatletter
\newcommand*\newnum[1]{%
  \ifcase\value{#1}%
    \or semel%
    \or bis%
    \or ter%
    \or quater%
    \or quinquies%
    % etc
    \else\@ctrerr
  \fi
}
\makeatother

% Don't use a space but use a thin space (\,) as Bernard did in his answer:    
\patchcmd\subequations
  {\def\theequation{\theparentequation\alph{equation}}}
  {\def\theequation{\theparentequation\ifnum\value{equation}<1 \else\protect\,\fi\newnum{equation}}}
  {}{\FAIL}

\begin{document}
    \begin{subequations}
        \begin{align}
            f(x) &= ax +b +c \\
            &= x \left(a + \frac{b}{x} + \frac{c}{x} \right) \\
            &= ax + x \left( \frac{b}{x} + \frac{c}{x} \right)
        \end{align}
    \end{subequations}
\end{document}

---

By the way: Your second question

Another question: how do you create a command of this type, but which doesn't need an argument? (For example, both \Alph and \Alph{} exist)

is not clear to me:

\documentclass{article}
\begin{document}
\show\Alph
\end{document}

reveals:

> \Alph=macro:
#1->\expandafter \@Alph \csname c@#1\endcsname .
l.3 \show\Alph

This means \Alph is a macro which in any case during its expansion does process a non-delimited argument.

That non-delimited argument must be the name of a LaTeX-counter.

That argument gets wrapped into \expandafter\@Alph\csname c@⟨counter⟩\endcsname.

The \expandafter ensures that the control-word-token \c@counter is formed from the \csname..\endcsname-construct before applying \@Alph to that control-word-token. The control-word-token \c@counter denotes the TeX-\count-register which underlies the LaTeX-counter in question.

So \Alph is a wrapper for \@Alph which serves the purpose of forming something (a control-word-token denoting a \count-register) that can be processed as a TeX-⟨number⟩-quantity from the name of a LaTeX-counter.

\@Alph in turn processes an argument which must be a TeX-⟨number⟩-quantity.

A TeX-⟨number⟩-quantity in turn can, e.g., be a control-word-token denoting a \count-register allocated in terms of \countdef.

But that's not all. A TeX-⟨number⟩-quantity can e.g., be an alphabetic constant, something like `\a or a sequence of digits.
(More details about the syntax rules for TeX-⟨number⟩-quantities can be found in the TeXBook, Chapter 24: Summary of Vertical Mode.)

Of course you can implement \newnum/\@newnum like this, too:

\documentclass{article}

\makeatletter
\newcommand*\@newnum[1]{%
  \ifcase#1%
    \or semel%
    \or bis%
    \or ter%
    \or quater%
    \or quinquies%
    % etc
    \else\@ctrerr
  \fi
}%
\newcommand*\newnum[1]{%
  \expandafter\@newnum\csname c@#1\endcsname
}%
\makeatother

\newcounter{testcounter}

\begin{document}

Testing \verb|\newnum| (argument must denote a \LaTeX-counter):
\medskip

\setcounter{testcounter}{1}
\newnum{testcounter}

\setcounter{testcounter}{2}
\newnum{testcounter}

\setcounter{testcounter}{3}
\newnum{testcounter}

\setcounter{testcounter}{4}
\newnum{testcounter}
\bigskip

Testing \verb|\@newnum| (argument must denote a \TeX-number-quantity):
\medskip

\makeatletter

\@newnum{1}%

\@tempcnta=2
\@newnum{\@tempcnta}%

\setcounter{testcounter}{3}
\@newnum{\value{testcounter}}%

\@newnum{`\^^D}

\makeatother

\end{document}

I couldn't patch it, but here is a code to redefine the environment (or define your own subequations environment):

\documentclass{article}
\usepackage{amsmath}
\usepackage{etoolbox}

\makeatletter%
\def\newnum#1{\expandafter\@newnum\csname c@#1\endcsname}
\def\@newnum#1{%
 \ifcase#1\or semel\or bis\or ter\or quater\or quinquies\or sexies\or septies\or octies\or nonies\else\@ctrerr\fi}
\renewenvironment{subequations}{%
  \refstepcounter{equation}%
  \protected@edef\theparentequation{\theequation}%
  \setcounter{parentequation}{\value{equation}}%
  \setcounter{equation}{0}%
  \def\theequation{\theparentequation\,\newnum{equation}}%
  \ignorespaces
}{%
  \setcounter{equation}{\value{parentequation}}%
  \ignorespacesafterend
}
\makeatother

\begin{document}
\setcounter{equation}{3}
\begin{equation}\label{eq}
  a = b
\end{equation}

 \begin{subequations}
 \begin{align}
 f(x) &= ax +b +c \\
 &= x \left(a + \frac{b}{x} + \frac{c}{x} \right) \\
 &= ax + x \left( \frac{b}{x} + \frac{c}{x} \right)
 \end{align}
 \end{subequations}
\end{document}