The missing vertical space in the inner center environment is because \@item uses \addvspace\@topsep to insert the space you want to have when leaving a blank line before \begin{center}, and \addvspace doesn't add this space when in vertical mode and \if@minipage is true—which is the case inside your mybox. So, in order to have this space inserted, you can set \@minipagefalse in the before upper option of your tcolorbox.
The vertical gap between the two boxes is essentially due to the default for /tcb/noparskip (see documentation of /tcb/autoparskip in the tcolorbox manual). If you set before=\par\noindent, you remove the added \smallskip:
\documentclass{article}
\usepackage{tcolorbox}
\makeatletter
\tcbset{mybox/.style={colback=white, colframe=blue, left=2mm, right=2mm,
fonttitle=\bfseries}, fontupper=\small,
before upper={\setlength{\parindent}{1em}%
\everypar{{\setbox0\lastbox}\@minipagefalse\everypar{}}},
before=\par\noindent, after=\par
}
\makeatother
\newtcolorbox{mybox}[1][]{mybox,#1}
\begin{document}
\begin{mybox}
This line should not be indented.
This line should be indented.
\begin{center}
This line should be centered. The spacing above is correct.
\end{center}
This line should be indented.
\begin{center}
This line should be centered. The spacing above is right.
\end{center}
This line should be indented.
\end{mybox}
\begin{mybox}
\section{Should not have gap above this heading}
\end{mybox}
\end{document}
If you really want to get rid of all the gap, suppress all vertical glue between the two boxes of interest here that are appended to the main vertical list. You can do this using \nointerlineskip, which is correctly used in vertical mode because of the after=\par we set:
\documentclass{article}
\usepackage{tcolorbox}
\makeatletter
\tcbset{mybox/.style={colback=white, colframe=blue, left=2mm, right=2mm,
fonttitle=\bfseries}, fontupper=\small,
before upper={\setlength{\parindent}{1em}%
\everypar{{\setbox0\lastbox}\@minipagefalse\everypar{}}},
before=\par\noindent, after=\par
}
\makeatother
\newtcolorbox{mybox}[1][]{mybox,#1}
\begin{document}
\begin{mybox}
This line should not be indented.
This line should be indented.
\begin{center}
This line should be centered. The spacing above is correct.
\end{center}
This line should be indented.
\begin{center}
This line should be centered. The spacing above is right.
\end{center}
This line should be indented.
\end{mybox}
\nointerlineskip
\begin{mybox}
\section{Should not have gap above this heading}
\end{mybox}
\end{document}
Note: I set the \@minipagefalse inside the temporary \everypar in order not to have extra vertical space inside the box if it starts with a center environment. But if you set it directly at the beginning of before upper, like this:
\tcbset{mybox/.style={colback=white, colframe=blue, left=2mm, right=2mm,
fonttitle=\bfseries}, fontupper=\small,
before upper={\@minipagefalse\setlength{\parindent}{1em}%
\everypar{{\setbox0\lastbox}\everypar{}}},
before=\par\noindent, after=\par
}
and start your box with a center environment:
\begin{mybox}
\begin{center}
This line should be centered. The spacing above is correct.
\end{center}
This line should be indented.
\begin{center}
This line should be centered. The spacing above is correct.
\end{center}
...
\end{mybox}
you'll get the extra space at the top of your box, like this:
