Thank you for noticing, I have changed 0 to -200. But it doesn't change nor the result of Nintegrate nor the result of plotting by Table. It's just "running", that's all.
plots = Table[
Plot[(1/( 2 Sqrt[Pi *((t) + 1)*(\[Epsilon])^(2)]) NIntegrate[
Exp[-(Abs[x - \[Xi]])^2/(4*((t) +
1)*(\[Epsilon])^(2))] g[\[Xi]] G[\[Xi]] (-1/(2*(\
\[Epsilon])^2)), {\[Xi], p1, p2}])/(1/(
2 Sqrt[Pi *((t) + 1)*(\[Epsilon])^(2)]) NIntegrate[
Exp[-(Abs[x - \[Xi]])^2/(4*((t) +
1)*(\[Epsilon])^(2))] g[\[Xi]], {\[Xi], p1, p2}]), {x,
0, 2}, PlotRange -> {-10, 10}], {t, -2, 0, .25}];
And
Plot3D[Evaluate[(1/(
2 Sqrt[Pi *((t) + 1)*(\[Epsilon])^(2)]) NIntegrate[
Exp[-(Abs[x - \[Xi]])^2/(4*((t) +
1)*(\[Epsilon])^(2))] g[\[Xi]] G[\[Xi]] (-1/(2*(\
\[Epsilon])^2)), {\[Xi], p1, p2}])/(1/(
2 Sqrt[Pi *((t) + 1)*(\[Epsilon])^(2)]) NIntegrate[
Exp[-(Abs[x - \[Xi]])^2/(4*((t) +
1)*(\[Epsilon])^(2))] g[\[Xi]], {\[Xi], p1, p2}])], {t,
0, 10}, {x, -10, 10}]
or even
Plot3D[(1/( 2 Sqrt[Pi *((t) + 1)*(\[Epsilon])^(2)]) NIntegrate[
Exp[-(Abs[x - \[Xi]])^2/(4*((t) +
1)*(\[Epsilon])^(2))] g[\[Xi]] G[\[Xi]] \
(-1/(2*(\[Epsilon])^2)), {\[Xi], p1, p2}])/(1/(
2 Sqrt[Pi *((t) + 1)*(\[Epsilon])^(2)]) NIntegrate[
Exp[-(Abs[x - \[Xi]])^2/(4*((t) +
1)*(\[Epsilon])^(2))] g[\[Xi]], {\[Xi], p1, p2}]), {t, 0,
10}, {x, -10, 10}]
is also useless.