Can I redefine a command to contain itself?

Important update

The LaTeX kernel from the 2020-10-01 release has \NewCommandCopy that does the same job of \LetLtxMacro and better. It also provides \ShowCommand to obtain the internal definition also of robust commands. So the code below should now be

\NewCommandCopy{\oldemph}{\emph}
\renewcommand{\emph}[1]{\textbf{\oldemph{#1}}}

The analysis below about the problems with \let is still valid.

Original answer

While something such as

\let\oldemph\emph
\renewcommand{\emph}[1]{\textbf{\oldemph{#1}}}

seems to work, it really doesn't, because \emph is defined with \DeclareRobustCommand. Where doesn't it work? suppose you write that code and then use \emph in a caption:

\caption{This is \emph{emphasized}}

what is written in the .aux file is

\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces This is \textbf  {\emph  {emphasized}}}}{1}}

This is because the original expansion of \emph is

\protect\emph§

(where with § I'm denoting a space in the name). So the \emph{emphasized} in the caption gets expanded successively into

\textbf{\oldemph{emphasized}}
\protect\textbf§{\protect\emph§{emphasized}}
\textbf§{\emph§{emphasized}

and written as

\textbf  {\emph  {emphasized}}

(because TeX adds a space after a command name when writing). In this case, when reading the file for building the List of Figures or Tables, TeX will find

\textbf{\textbf{\oldemph{emphasized}}}

which is innocuous. But it's not very difficult to imagine situations where this can lead to infinite loops.

The package letltxmacro prevents this problem:

\usepackage{letltxmacro}
\LetLtxMacro{\oldemph}{\emph}
\renewcommand{\emph}[1]{\textbf{\oldemph{#1}}}

would be a safer way to proceed.

Never use \let on commands defined with \DeclareRobustCommand or commands defined with \newcommand and having an optional argument.

The former can be recognized by the fact that \show\command returns

> \command=macro:
->\protect \command  .

The latter are distinguished by the fact that \show\command returns

> \command=macro:
->\@protected@testopt \command \\command {...}.

(where ... is actually the default for the optional argument).

Yes, this is possible using \g@addto@macro or some expanded definition (using \expandafter). Here are some examples:

\documentclass{article}
\setlength{\parindent}{0pt}% Just for this example
\begin{document}
\newcommand{\somecmd}{some \textbf{text}}
\somecmd \par
\makeatletter
\g@addto@macro\somecmd{ and some \textit{more} text}
\makeatother
\somecmd

\renewcommand{\somecmd}{some \textbf{text}}
\somecmd \par
\makeatletter
\expandafter\def\expandafter\somecmd\expandafter{\somecmd{} and some \textit{more} text}
\makeatother
\somecmd
\end{document}

The latter example, just like the former, appends contents to a macro. The placement of \somecmd in the latter could be used to prepend something.

However, nothing prevents you from making a new command that includes the old command as part of its expansion, since memory problems tend to be a thing of the past.

You run into problems when the contents of the newly-defined macro does not expand properly. Using your case as an example: although \somecommand may expand as expected, the center environment does not.

Sometimes, you need to use the original name of the command, and you cannot use an environment because of some undesirable side effects. I have the case with \LettrineFontHook which is not directly called in my code. Here, the only solution I have found is to use \let to store and restore the command:

\let\tempLettrineFontHook\LettrineFontHook
\renewcommand{\LettrineFontHook}{\fontseries{bx}}
\lettrine{F}{}
\let\LettrineFontHook\tempLettrineFontHook

If I try to define an environment like this:

\newenvironment{boldLettrine}{%
\renewcommand{\LettrineFontHook}{\fontseries{bx}}%
}{}
% let's use this environment in the document now:
\begin{boldLettrine}
\lettrine{F}{}
\end{boldLettrine}

For reasons I ignore, the indentation of the lettrine is broken.

EDIT: From egreg remark, in your example, this method cannot apply since the new command is defined with the help of the original one, leading to possible infinite loops. So I cancel the application to your use case.

Append to a defined command

If you want to simply append something to a command after it, you can use the LaTeX kernel \g@addto@macro:

\def\authors{}
\def\addauthor#1{%
    \g@addto@macro\authors{
        \begin{tabular}[t]{c}%
            #1
        \end{tabular}%
    }
}

Then in the preamble you can:

\addauthor{Cristiano Ronaldo}
\addauthor{Lionel Messi}

The \authors will contain a printable list of authors that you can use in your maketitle, for example.

Append and Prepend to a defined command

I have not test it, but try:

\def\aroundto#1#2#3{\toks0={#2}\toks1=\expandafter{#1}\toks2={#3}\edef#1{\the\toks0 \the\toks1 \the\toks2}}

References: both solutions were found in The Hacks blog.