edit adds a patch to another, pure amsmath, problem; see bottom of answer
Not an easy one. First of all indeed hyperref should always be loaded after amsmath because it does things depending on whether amsmath is loaded, but does not delay this to At Begin Document.
The problem of a unary minus is already taken care of by hyperref's redefinition of \equation (code comment near lines 7553 of hyperref.pdf)
\mathopen is needed in case the equation starts with an unary minus, for example.
but this redefinition is done only if amsmath is not loaded. In the case hyperref detects amsmath it modifies the amsmath definition in a different way, moving the \refstepcounter and ensuing hyperlink business to the start of the formula, to avoid putting things in the surrounding vertical list. However it appears that hyperref forgot in that case to also add the \mathopen thing.
Here is a proposed patch.
not tested in any thorough way it would be better to get hyperref to patch only equation etc...
\documentclass{article}
\usepackage{amsmath}
% always load hyperref after amsmath
\usepackage{hyperref}
\makeatletter
\def\hyper@refstepcounter#1{%
\edef\This@name{#1}%
\ifx\This@name\name@of@eq
\@ifundefined{theHequation}{%
\make@stripped@name{\theequation}%
\let\theHequation\newname
}{}%
\fi
\HyCnt@ProvideTheHCounter{#1}%
\hyper@makecurrent{#1}%
\ifmeasuring@
\else
%% \ifmmode\mathopen\bgroup\fi %% first version of the patch (line I)
\Hy@raisedlink{%
\hyper@anchorstart{\@currentHref}\hyper@anchorend
}%
%% \ifmmode\egroup\fi %% first version of the patch (line II)
\fi
\ifmmode\mathopen{}\fi %% second version: THIS LINE ADDED IN PATCH
}
\makeatother
\begin{document}
\begin{equation}
-1 = -1 % output is now OK
\end{equation}
\end{document}

A similar problem non-related to hyperref exists in amsmath, with the \multline environment, as pointed out in comments. I can propose the following patch, with no guarantee as this is a job for someone who knows all the innards of amsmath already.
The patches are done with the help of etoolbox.
about this patch I am not sure why amsmath has \displaystyle{} and not \displaystyle alone which would not create the problem; an alternative patch would be to simply replace \displaystyle{} by \displaystyle, but there might be some reason for the \displaystyle{} (compare the \shoveleft definition in amsmath.sty).
\documentclass{article}
\usepackage{amsmath}
% un deuxième problème a émergé qui n'est pas lié à hyperref.
\usepackage{etoolbox}
\begin{document}\thispagestyle{empty}
Before the patch:
\begin{multline}
-1 = -1\\ -2 = -2
\end{multline}
\makeatletter
\patchcmd
{\multline@}{\displaystyle{}}{\displaystyle{}\mathopen{}}
{\typeout{multpatch SUCCESS}}
{\typeout{multpatch FAILURE}}
\patchcmd
{\mmeasure@}{\displaystyle{}}{\displaystyle{}\mathopen{}}
{\typeout{mmpatch SUCCESS}}
{\typeout{mmpatch FAILURE}}
\makeatother
After the patch
\begin{multline}
-1 = -1\\ -2 = -2
\end{multline}
\end{document}

hyperrefbug. – Mar 29 '15 at 14:19amsmathbug with themultlineenvironment, which manifests itself as in yourMWE, although it does not have the same cause (being present withouthyperref). See my answer for proposed patches. – Mar 29 '15 at 15:24