The xtick and xmin and xmax do not affect each other, as a result, just because you have xtick going down to -7 won't make xmin go down that far either.
Now, regarding placing the ticks above the axis, this can be easily done for all of them as shown in the example below. If, however, you want to have some ticks above and some ticks below, then I'm not sure how to do it. As an alternative, you could have the ticks slightly offset so they don't overlap with the lines.
Negative Values, Ticks Above
\documentclass[border={10pt}, convert]{standalone}
\usepackage{amsmath}
\usepackage{tikz}
\usepackage{pgfplots}
\pgfplotsset{compat=1.13}
\begin{filecontents}{data.dat}
n xn
-5 -0.0
-4 -0.0
-3 -3.0
-2 -2.0
-1 -1.0
0 -0.0
1 -1.0
2 -2.0
3 -3.0
4 -0.0
5 -0.0
\end{filecontents}
\begin{document}
\begin{tikzpicture}
\begin{axis}[
axis lines=middle,
xlabel={$n$},
ylabel={$\boldsymbol{x[n]}$},
ylabel style={
at={(yticklabel cs: 0.1, 5pt)},
},
xtick={-6, -5, ..., 6},
xmin=-7, xmax=7,
ymin=-5, ymax=1,
xticklabel style={
anchor=south,
inner sep=5pt,
}
]
\addplot [ycomb, black, thick, mark=*] table [x={n}, y={xn}] {data.dat};
\end{axis}
\end{tikzpicture}
\end{document}
Pos/Neg Values, Ticks Offset
\documentclass[border={10pt}, convert]{standalone}
\usepackage{amsmath}
\usepackage{tikz}
\usepackage{pgfplots}
\pgfplotsset{compat=1.13}
\begin{filecontents}{data.dat}
n xn
-5 -0.0
-4 -0.0
-3 -3.0
-2 -2.0
-1 -1.0
0 -0.0
1 1.0
2 2.0
3 3.0
4 0.0
5 0.0
\end{filecontents}
\begin{document}
\begin{tikzpicture}
\begin{axis}[
axis lines=middle,
xlabel={$n$},
ylabel={$\boldsymbol{x[n]}$},
% xtick={-6, -5, ..., 6},
xmin=-7, xmax=7,
ymin=-5, ymax=5,
xticklabel style={
anchor=north east,
inner sep=2pt,
}
]
\addplot [ycomb, black, thick, mark=*] table [x={n}, y={xn}] {data.dat};
\end{axis}
\end{tikzpicture}
\end{document}
Tick Label Background
As suggested by Rman, another option is to add a background to the ticks so that they are visible over the graphics. He also mentions the use of enlarge x limits which is a way of making xmin and xmax bigger. In particular, enlarge x limits={abs=1.5} will automatically calculate xmin and xmax, and then add -1.5 and 1.5 units respectively to each value.
Since I don't want a completely white box, I have opted to use a translucent white background so that the line can still be seen underneath. Also note that axis on top is required; otherwise, the ticks end up underneath.
\documentclass[border={10pt}, convert]{standalone}
\usepackage{amsmath}
\usepackage{tikz}
\usepackage{pgfplots}
\pgfplotsset{compat=1.13}
\begin{filecontents}{data.dat}
n xn
-5 -0.0
-4 -0.0
-3 -3.0
-2 -2.0
-1 -1.0
0 -0.0
1 1.0
2 2.0
3 3.0
4 0.0
5 0.0
\end{filecontents}
\begin{document}
\begin{tikzpicture}
\begin{axis}[
axis lines=middle,
xlabel={$n$},
ylabel={$\boldsymbol{x[n]}$},
% xtick={-6, -5, ..., 6},
enlarge x limits={abs=1.5},
ymin=-5, ymax=5,
xticklabel style={
fill=white,
fill opacity=0.7,
text opacity=1,
inner sep=1pt,
},
axis on top,
]
\addplot [ycomb, black, thick, mark=*] table [x={n}, y={xn}] {data.dat};
\end{axis}
\end{tikzpicture}
\end{document}
xminandxmaxto set the limits explicitly, just like you did for the limits of the y axis. The second part is harder. Do you want to place the tick (and ticklabel I guess) above the axis only in case of a negative signalvalue? I don't know if/how that can be done.xminandxmax, you can also setenlarge x limit={abs=0.5}which is what I suspect you want. See tex.stackexchange.com/questions/294398/…