How can I show that a sphere is rolling in a simulation?

Any reason why you want to show a Sphere? To my mind using another shape would be a lot more fun. Here's m_goldberg's code with a demo from SphericalPlot3D:

myWheel = 
  SphericalPlot3D[
   .8 + Sin[5 ϕ]/5, {θ, 0, Pi}, {ϕ, 0, 2 Pi}, 
   PlotStyle -> Directive[Orange, Opacity[.9], Specularity[White, 10]], 
   Mesh -> None, PlotPoints -> 30
   ];

frames =
  Table[
   Graphics3D[
    Rotate[
     Translate[
      myWheel[[1]],
      {location[t], R, R}
      ],
     zhuanjiao[t],
     {0, 1, 0},
     (*changed to get rolling to work*)
     {location[t], R, R}
     ],
    Axes -> True,
    AxesEdge -> {{1, -1}, {-1, -1}, {-1, 1}},
    Ticks -> None,
    Boxed -> False,
    Lighting -> "Neutral",
    PlotRange -> {{0, 10}, {0, 2 R}, {0, 2 R}},
    AxesLabel -> {x, y, z},
    ViewPoint -> 4*{1, -2, .5}
    ],
   {t, 0, 1, .05}
   ];

rast = Rasterize[#, ImageResolution -> 144] & /@ frames;

CloudExport[rast, "GIF", "test.gif", "AnimationRepetitions" -> Infinity, Permissions -> "Public"]

Well, you might do it by putting spots on the sphere. However, if you were to do that, you would the rotation you have imposed is not the rotation of a rolling ball. At least, it doesn't look it to me.

The following demonstrates a set of points, located on the surface of a ball, being moved and rotated by your functions location and zhuanjiao. Evaluate it to see what I mean.

R = 1;

v0 = 10; w0 = 8; mu = 0.2; g = 9.8;
δ = (2 (v0 + w0 R))/(5 mu g);

location[t_] := 
  Piecewise[
    {{If[t <= δ, 
        v0 t - 1/2* mu*g*t^2 + R, 
        (3 v0 - 2 w0 R)/5 (t - δ) + v0 δ - 1/2* mu*g*δ^2 + R], 
      3 v0 - 2 w0 R < 0}, (*<0 roll back*)
      {If[t <= δ, 
         v0 t - 1/2* mu*g*t^2 + R, 
         v0 δ - 1/2* mu*g*δ^2 + R], 
      3 v0 - 2 w0 R == 0},
      {If[t <= δ, 
         v0 t - 1/2* mu*g*t^2 + R, 
         (3 v0 - 2 w0 R)/5 (t - δ) + v0 δ - 1/2* mu*g*δ^2 + R], 
      3 v0 - 2 w0 R > 0}}]

zhuanjiao[t_] := 
  If[t <= δ, 
    w0 t - (3 mu g)/(4 R) t^2, 
    w0 δ - (3 mu g)/(4 R) δ^2 - (3 v0 - 2 w0 R)/(5 R) (t - δ)]

Manipulate[
  Graphics3D[
    Rotate[
      Translate[Point[SpherePoints[100]], {location[t], R, R}],
      -zhuanjiao[t],
      {1, 0, 0}, (* changed to get rolling to look right *)
      {location[t], R, R}],
    Axes -> True,
    AxesOrigin -> {0, 0, 0},
    PlotRange -> {{0, 10}, {0, 2 R}, {0, 2 R}},
    AxesLabel -> {x, y, z}],
  {t, 0, 1, .01}]