Help with Help Lines

Question

I am plotting lattice paths to two points. I would like to have help lines drawn on top of the Cartesian plane so that any line that is an integral distance from an axis is a solid line and any line that is an odd integral multiple of 0.5 from an axis is dashed. The help lines that I have are not even consistent with the axes!

How to I get the O, A, and B typeset over the help lines. The option [fill=white,circle,inner sep=0pt] is ignored in my code. So, one of the (misplaced) help lines is drawn over it.

\documentclass{amsart}
\usepackage{tikz}
\usetikzlibrary{calc,angles,positioning}



\usepackage{pgfplots}
\pgfplotsset{compat=1.11}

\begin{document}

\hspace*{\fill}
\begin{tikzpicture}
\begin{axis}[
    xmin=-11,xmax=11,ymin=-11,ymax=11,
    axis lines=middle,
    enlargelimits={abs=0.5},
    axis line style={latex-latex},
    ticklabel style={font=\tiny,fill=white},
    xlabel style={at={(ticklabel* cs:1)},anchor=north west},
    ylabel style={at={(ticklabel* cs:1)},anchor=south west}
        ]

\coordinate (O) at (0,0);
\node[fill=white,circle,inner sep=0pt] (O-label) at ($(O)+(-135:10pt)$) {$O$};
\coordinate (A) at (-2.5,9.5);
\node[fill=white,circle] (A-label) at ($(A)+(-180:10pt)$) {$A$};
\coordinate (B) at (7,-3.5);
\node[fill=white,circle] (B-label) at ($(B)+(-45:10pt)$) {$B$};

\draw[fill] (O) circle (1.5pt);
\draw[fill] (A) circle (1.5pt);
\draw[fill] (B) circle (1.5pt);

%These commands are for drawing a path from O to A.
\coordinate (A1) at (-0.5,0);
\coordinate (A2) at (-.5,3);
\coordinate (A3) at (-2.5,3);

\draw[line width=1pt] (O) -- (A1) -- (A2) -- (A3) -- (A);

%These commands are for drawing a path from O to B.
\coordinate (B1) at (2,0);
\coordinate (B2) at (2,-2.5);
\coordinate (B3) at (5,-2.5);
\coordinate (B4) at (5,-3.5);
\coordinate (B) at (7,-3.5);

\draw[line width=1pt] (O) -- (B1) -- (B2) -- (B3) -- (B4) -- (B);

\draw[help lines,line width=0.1pt,blue] (-10.5, -10.5) grid (10.5,10.5);


\end{axis}
\end{tikzpicture}
\hspace{\fill}
\vskip0.25in

\end{document}

Answer 1

^{This is also an answer to the comment for another answer of another question.}

To get help lines overprinted, they have to be printed first below the following objects. The \draw command should be placed before the other elements.

The next problem are the misplaced lines. The following command adds the grid lines at least in the area (-10, -10) to (10, 10):

\draw[
  help lines,
  line width=0.1pt,
  blue,
  shift={($(1, 1) - (0, 0)$)},
] (-11, -11) grid[step={($(5, 5) - (0, 0)$)}] (9, 9);

Full example:

\documentclass{amsart}
\usepackage{tikz}
\usetikzlibrary{calc,angles,positioning}
\usepackage{pgfplots}
\pgfplotsset{compat=1.11}

\begin{document}   
\begin{tikzpicture}
\begin{axis}[
    xmin=-11,xmax=11,ymin=-11,ymax=11,
    axis lines=middle,
    enlargelimits={abs=0.5},
    axis line style={latex-latex},
    ticklabel style={font=\tiny,fill=white},
    xlabel style={at={(ticklabel* cs:1)},anchor=north west},
    ylabel style={at={(ticklabel* cs:1)},anchor=south west}
        ]

\draw[
  help lines,
  line width=0.1pt,
  blue,
  shift={($(1, 1) - (0, 0)$)},
] (-11, -11) grid[step={($(5, 5) - (0, 0)$)}] (9, 9);

\coordinate (O) at (0,0);
\node[fill=white,circle,inner sep=0pt] (O-label) at ($(O)+(-135:10pt)$) {$O$};
\coordinate (A) at (-2.5,9.5);
\node[fill=white,circle] (A-label) at ($(A)+(-180:10pt)$) {$A$};
\coordinate (B) at (7,-3.5);
\node[fill=white,circle] (B-label) at ($(B)+(-45:10pt)$) {$B$};

\draw[fill] (O) circle (1.5pt);
\draw[fill] (A) circle (1.5pt);
\draw[fill] (B) circle (1.5pt);

%These commands are for drawing a path from O to A.
\coordinate (A1) at (-0.5,0);
\coordinate (A2) at (-.5,3); 
\coordinate (A3) at (-2.5,3);

\draw[line width=1pt] (O) -- (A1) -- (A2) -- (A3) -- (A);

%These commands are for drawing a path from O to B.
\coordinate (B1) at (2,0);   
\coordinate (B2) at (2,-2.5);
\coordinate (B3) at (5,-2.5);
\coordinate (B4) at (5,-3.5);
\coordinate (B) at (7,-3.5);

\draw[line width=1pt] (O) -- (B1) -- (B2) -- (B3) -- (B4) -- (B);

\end{axis}
\end{tikzpicture}
\end{document}   

The full plot area from -11 to 11 in both directions can be covered by:

\begin{scope}
  \clip (-11, -11) rectangle (11, 11);
  \draw[
    help lines,
    line width=0.1pt,
    blue,
    shift={($(1, 1) - (0, 0)$)},
  ] (-16, -16) grid[step={($(5, 5) - (0, 0)$)}] (10, 10);
\end{scope}

However, the axis tick labels are overwritten by the help lines. Therefore I would rather use the grid option of the axis environment:

\begin{axis}[
  ...,
  grid,
  grid options={line width=.1pt, draw=gray!30},
]
  ...
\end{axis}