nccmath affecting minipage widths

It appears that nccmath indeed breaks minipages containing only a math display, setting them to have the width resulting from adding the space on the left of the display and the equation width.

If the minipage contains "paragraph material", its width is correctly set.

Solution: start a minipage that should contain only a display with

\hrule height 0pt width \hsize

Some examples (in the code the \hrule has a height in order to make it visible)

\documentclass[a4paper,fleqn]{article}

\usepackage{amsmath}
\usepackage{nccmath}% Needed for fleqn environment

\begin{document}
% let's have a "ruler"
\hbox to\hsize{\leaders\hrule\hskip0pt plus 7fill
  \kern-.2pt\vrule depth 2pt\kern-.2pt
  \leaders\hrule\hskip0pt plus 3fill}
\medskip

Example 1: we put a rule with the desired width

\noindent\begin{minipage}{0.70\columnwidth}
\hrule width \hsize\kern-.4pt
\begin{align*}
    y &= x 
\end{align*}
\end{minipage}%
\begin{minipage}{0.30\columnwidth}
\hrule
    Note: abcd

Where should this go?
\hrule
\end{minipage}

\medskip

Example 2: we don't specify the rule's width

\noindent\begin{minipage}{0.70\columnwidth}
\hrule\kern-.4pt
\begin{align*}
    y &= x 
\end{align*}
\end{minipage}%
\begin{minipage}{0.30\columnwidth}
\hrule
    Note: abcd

Where should this go?
\hrule
\end{minipage}

\medskip

Example 3: text before the display

\noindent\begin{minipage}{0.70\columnwidth}
\hrule\kern-.4pt
Some text before
\begin{align*}
    y &= x 
\end{align*}
\end{minipage}%
\begin{minipage}{0.30\columnwidth}
\hrule
    Note: abcd

Where should this go?
\hrule
\end{minipage}

\medskip

Example 4: text below the display

\noindent\begin{minipage}{0.70\columnwidth}
\hrule\kern-.4pt
\begin{align*}
    y &= x 
\end{align*}
Some text below
\end{minipage}%
\begin{minipage}{0.30\columnwidth}
\hrule
    Note: abcd

Where should this go?
\hrule
\end{minipage}

\end{document}

In the comment within your code I see you need nccmath for the fleqn environment. You can define that environment yourself in your preamble without loading nccmath. You can use the code from that package:

\makeatletter
\newenvironment*{fleqn}[1][\z@]{\@fleqntrue
  \setlength\@mathmargin{#1}\ignorespaces
}{%
  \ignorespacesafterend
}
\makeatother