avoid overlapping of labels and arcs in Tikz

The auto option is not magic. Its effect depends on the current direction, so it cannot take surrounding drawings into effect. You will have to interfer manually, but there are ways to make this feel kind of automatic. The easiest way to fix this is to use an angle for the bend right style:

\documentclass{standalone}

\usepackage{tikz}
\begin{document}
\usetikzlibrary{automata,chains}
\begin{tikzpicture}[start chain=going right]
\node[state,on chain] (-2) {-2};
\node[state,on chain] (-1) {-1};
\node[state,on chain] (0) {0};
\node[draw=red,fill=red,state,on chain] (+1) {+1};
\node[draw=red,fill=red,state,on chain] (+2) {+2};

\draw[->] (-1) to node[auto] {$P(0)$}(-2);
\path[->] (-1) edge  [loop above] node[auto] {$P(1)$} ();
\draw[->] (-1) to node[auto] {$P(2)$}(0);
\draw[->] (-1) to[bend right=60] node[auto] {$P(3)$}(+1);
\draw[->] (-1) to[bend right=60] node[auto] {$P(4)$}(+2);
%                           ^^^ change is here

\draw[->] (+1) to node[above]{$1$} (0);
\draw[->] (+2) to[bend right] node[above]{$1$} (0);
\end{tikzpicture}
\end{document}

This yields:

But i would suggest the following, for consistency (also, i like explicit positioning, but that is a matter of taste):

\draw[->] (-1) to[bend right=30] node[below] {$P(3)$}(+1);
\draw[->] (-1) to[bend right=50] node[below] {$P(4)$}(+2);

This yields

Use the auto option globally and add swap to those nodes that end up on the wrong side.

\documentclass[border=2mm]{standalone}
\usepackage{tikz}
\usetikzlibrary{automata,chains}
\begin{document}
\begin{tikzpicture}[start chain=going right,auto]
\node[state,on chain] (-2) {-2};
\node[state,on chain] (-1) {-1};
\node[state,on chain] (0) {0};
\node[draw=red,fill=red,state,on chain] (+1) {+1};
\node[draw=red,fill=red,state,on chain] (+2) {+2};

\draw[->] (-1) to node[swap] {$P(0)$}(-2);
\path[->] (-1) edge  [loop above] node {$P(1)$} ();
\draw[->] (-1) to node {$P(2)$}(0);
\draw[->] (-1) to[bend right] node[swap] {$P(3)$}(+1);
\draw[->] (-1) to[bend right=50] node[swap] {$P(4)$}(+2);

\draw[->] (+1) to node[swap]{$1$} (0);
\draw[->] (+2) to[bend right] node[swap]{$1$} (0);
\end{tikzpicture}
\end{document}

For exercise: how to make OP code shorter and without overlapping of edges labels. With use of quotes library, and predefined edge angle, and reduced labels' font size to \footnotesize:

\documentclass[tikz, margin=3mm]{standalone}
\usetikzlibrary{automata, chains, quotes}

\begin{document}
    \begin{tikzpicture}[auto,
     bend angle = 45,
    start chain = going right,
every state/.append style = {on chain},
every edge quotes/.style = {inner sep=2pt, font=\footnotesize}
                        ]
\node[state] (-2) {$-2$};
\node[state] (-1) {$-1$};
\node[state] (0)   {0};
\node[state,draw=red,fill=red] (+1) {$+1$};
\node[state,draw=red,fill=red] (+2) {$+2$};

\draw[->]   (-1) edge["$P(0)$"] (-2)
            (-1) edge[loop above,  "$P(1)$"] ()
            (-1) edge["$P(2)$"] (0)
            (-1) edge[bend right,"$P(3)$"] (+1)
            (-1) edge[bend right,"$P(4)$"] (+2)%change is here
            (+1) edge["$1$"]    (0)
            (+2) edge[bend right,"$1$"] (0);
\end{tikzpicture}
\end{document}

the image of the automaton become: