Use
Manipulate[mcPlot[n], {n, 1, 10, 1}]
to avoid non-integer n being passed to RandomReal[{0, π/2}, n].
For the second part of the question, you can use Grid instead of GraphicsGrid:
ClearAll[mcPlot]
mcPlot[nn_] := Module[{x, y, n = nn, colorFunction},
x = RandomReal[{0, π/2}, n]; y = N[Sin[x]];
colorFunction = ColorData["BrightBands"];
text = Grid[{{Style[TraditionalForm[
HoldForm[Divide[1, "n"] Sum[Sin[Subscript["x", i]], {i, 1, "n"}]]],
Large, Bold, RGBColor[0.2, 0.2, 0.2]],
Style["==", Large, Bold, RGBColor[0.2, 0.2, 0.2]],
Style[N[Mean[y]Style[Round[Mean[y], 5].00001], Large, Bold, RGBColor[0.2, 0.2, 0.2]]},
{Style[TraditionalForm[
HoldForm[Divide[Abs[s - Subscript[s, 0]], Subscript[s, 0]]]],
Large, Bold, RGBColor[0.2, 0.2, 0.2]],
Style["==", Large, Bold, RGBColor[0.2, 0.2, 0.2]],
Style[N[Abs[Mean[y]Style[Round[Abs[Mean[y] π/2 - 1], 5].00001], Large, Bold, RGBColor[0.2, 0.2, 0.2]]}}];
Row[{Show[Plot[Sin[x], {x, 0, π/2}, Frame -> True, ImageSize -> Large,
PlotStyle -> Thickness[0.01],
FrameLabel -> {{Text[Style["Sin[x]", Medium, Bold, RGBColor[0.2, 0.5, 0.9]]],
None},
{Text[ Style["x ∈ [0, \!\(\*FractionBox[\(π\), \\(2\)]\)]",
Medium, Bold, RGBColor[0.2, 0.5, 0.9]]], None}}],
Graphics[{EdgeForm[Opacity[.5]], {colorFunction[#[[1]]/(π/2)],
Opacity[.05], Rectangle[{0, 0}, {#[[1]], #[[2]]}],
colorFunction[#[[1]]/(π/2)], Opacity[.05],
Rectangle[{#[[1]], #[[2]]}, {π/2, 0}]} & /@ Transpose[{x, y}],
{PointSize[0.015], Point[Transpose[{x, y}]]}}]], text}, Spacer[10]]]
Manipulate[mcPlot[n], {n, 1, 10, 1}]
